diff --git a/doc/source/whatsnew/v2.2.0.rst b/doc/source/whatsnew/v2.2.0.rst
index 09067d541d01b..3bd817b6cb1f0 100644
--- a/doc/source/whatsnew/v2.2.0.rst
+++ b/doc/source/whatsnew/v2.2.0.rst
@@ -233,6 +233,7 @@ For example:
 Other Deprecations
 ^^^^^^^^^^^^^^^^^^
 - Changed :meth:`Timedelta.resolution_string` to return ``h``, ``min``, ``s``, ``ms``, ``us``, and ``ns`` instead of ``H``, ``T``, ``S``, ``L``, ``U``, and ``N``, for compatibility with respective deprecations in frequency aliases (:issue:`52536`)
+- Deprecated :meth:`Index.format`, use ``index.astype(str)`` or ``index.map(formatter)`` instead (:issue:`55413`)
 - Deprecated allowing non-keyword arguments in :meth:`DataFrame.to_clipboard`. (:issue:`54229`)
 - Deprecated allowing non-keyword arguments in :meth:`DataFrame.to_csv` except ``path_or_buf``. (:issue:`54229`)
 - Deprecated allowing non-keyword arguments in :meth:`DataFrame.to_dict`. (:issue:`54229`)
@@ -260,6 +261,7 @@ Other Deprecations
 - Deprecated the extension test classes ``BaseNoReduceTests``, ``BaseBooleanReduceTests``, and ``BaseNumericReduceTests``, use ``BaseReduceTests`` instead (:issue:`54663`)
 - Deprecated the option ``mode.data_manager`` and the ``ArrayManager``; only the ``BlockManager`` will be available in future versions (:issue:`55043`)
 - Deprecating downcasting the results of :meth:`DataFrame.fillna`, :meth:`Series.fillna`, :meth:`DataFrame.ffill`, :meth:`Series.ffill`, :meth:`DataFrame.bfill`, :meth:`Series.bfill` in object-dtype cases. To opt in to the future version, use ``pd.set_option("future.no_silent_downcasting", True)`` (:issue:`54261`)
+-
 
 .. ---------------------------------------------------------------------------
 .. _whatsnew_220.performance:
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index c3cab965041e0..e8ea4729e8232 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -1370,6 +1370,14 @@ def format(
         """
         Render a string representation of the Index.
         """
+        warnings.warn(
+            # GH#55413
+            f"{type(self).__name__}.format is deprecated and will be removed "
+            "in a future version. Convert using index.astype(str) or "
+            "index.map(formatter) instead.",
+            FutureWarning,
+            stacklevel=find_stack_level(),
+        )
         header = []
         if name:
             header.append(
@@ -1383,6 +1391,31 @@ def format(
 
         return self._format_with_header(header=header, na_rep=na_rep)
 
+    _default_na_rep = "NaN"
+
+    @final
+    def _format_flat(
+        self,
+        *,
+        include_name: bool,
+        formatter: Callable | None = None,
+    ) -> list[str_t]:
+        """
+        Render a string representation of the Index.
+        """
+        header = []
+        if include_name:
+            header.append(
+                pprint_thing(self.name, escape_chars=("\t", "\r", "\n"))
+                if self.name is not None
+                else ""
+            )
+
+        if formatter is not None:
+            return header + list(self.map(formatter))
+
+        return self._format_with_header(header=header, na_rep=self._default_na_rep)
+
     def _format_with_header(self, *, header: list[str_t], na_rep: str_t) -> list[str_t]:
         from pandas.io.formats.format import format_array
 
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 94ad556219b35..42af9556e04e1 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -14,6 +14,7 @@
     cast,
     final,
 )
+import warnings
 
 import numpy as np
 
@@ -42,6 +43,7 @@
     cache_readonly,
     doc,
 )
+from pandas.util._exceptions import find_stack_level
 
 from pandas.core.dtypes.common import (
     is_integer,
@@ -187,6 +189,7 @@ def _convert_tolerance(self, tolerance, target):
 
     # --------------------------------------------------------------------
     # Rendering Methods
+    _default_na_rep = "NaT"
 
     def format(
         self,
@@ -198,6 +201,14 @@ def format(
         """
         Render a string representation of the Index.
         """
+        warnings.warn(
+            # GH#55413
+            f"{type(self).__name__}.format is deprecated and will be removed "
+            "in a future version. Convert using index.astype(str) or "
+            "index.map(formatter) instead.",
+            FutureWarning,
+            stacklevel=find_stack_level(),
+        )
         header = []
         if name:
             header.append(
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index c5cab225fa7b1..3ef3cb6d56ed1 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -1383,6 +1383,15 @@ def format(
         sparsify=None,
         adjoin: bool = True,
     ) -> list:
+        warnings.warn(
+            # GH#55413
+            f"{type(self).__name__}.format is deprecated and will be removed "
+            "in a future version. Convert using index.astype(str) or "
+            "index.map(formatter) instead.",
+            FutureWarning,
+            stacklevel=find_stack_level(),
+        )
+
         if name is not None:
             names = name
 
@@ -1445,6 +1454,69 @@ def format(
         else:
             return result_levels
 
+    def _format_multi(
+        self,
+        *,
+        include_names: bool,
+        sparsify: bool | None | lib.NoDefault,
+        formatter: Callable | None = None,
+    ) -> list:
+        if len(self) == 0:
+            return []
+
+        stringified_levels = []
+        for lev, level_codes in zip(self.levels, self.codes):
+            na = _get_na_rep(lev.dtype)
+
+            if len(lev) > 0:
+                taken = formatted = lev.take(level_codes)
+                formatted = taken._format_flat(include_name=False, formatter=formatter)
+
+                # we have some NA
+                mask = level_codes == -1
+                if mask.any():
+                    formatted = np.array(formatted, dtype=object)
+                    formatted[mask] = na
+                    formatted = formatted.tolist()
+
+            else:
+                # weird all NA case
+                formatted = [
+                    pprint_thing(na if isna(x) else x, escape_chars=("\t", "\r", "\n"))
+                    for x in algos.take_nd(lev._values, level_codes)
+                ]
+            stringified_levels.append(formatted)
+
+        result_levels = []
+        for lev, lev_name in zip(stringified_levels, self.names):
+            level = []
+
+            if include_names:
+                level.append(
+                    pprint_thing(lev_name, escape_chars=("\t", "\r", "\n"))
+                    if lev_name is not None
+                    else ""
+                )
+
+            level.extend(np.array(lev, dtype=object))
+            result_levels.append(level)
+
+        if sparsify is None:
+            sparsify = get_option("display.multi_sparse")
+
+        if sparsify:
+            sentinel: Literal[""] | bool | lib.NoDefault = ""
+            # GH3547 use value of sparsify as sentinel if it's "Falsey"
+            assert isinstance(sparsify, bool) or sparsify is lib.no_default
+            if sparsify is lib.no_default:
+                sentinel = sparsify
+            # little bit of a kludge job for #1217
+            result_levels = sparsify_labels(
+                result_levels, start=int(include_names), sentinel=sentinel
+            )
+
+        return result_levels
+
     # --------------------------------------------------------------------
     # Names Methods
 
diff --git a/pandas/io/formats/excel.py b/pandas/io/formats/excel.py
index b344d9849f16c..684cd4340cd2b 100644
--- a/pandas/io/formats/excel.py
+++ b/pandas/io/formats/excel.py
@@ -623,8 +623,8 @@ def _format_header_mi(self) -> Iterable[ExcelCell]:
             return
 
         columns = self.columns
-        level_strs = columns.format(
-            sparsify=self.merge_cells, adjoin=False, names=False
+        level_strs = columns._format_multi(
+            sparsify=self.merge_cells, include_names=False
         )
         level_lengths = get_level_lengths(level_strs)
         coloffset = 0
@@ -813,8 +813,8 @@ def _format_hierarchical_rows(self) -> Iterable[ExcelCell]:
 
             if self.merge_cells:
                 # Format hierarchical rows as merged cells.
-                level_strs = self.df.index.format(
-                    sparsify=True, adjoin=False, names=False
+                level_strs = self.df.index._format_multi(
+                    sparsify=True, include_names=False
                 )
                 level_lengths = get_level_lengths(level_strs)
 
diff --git a/pandas/io/formats/format.py b/pandas/io/formats/format.py
index cac83e2a48972..9d75f4cf3ab49 100644
--- a/pandas/io/formats/format.py
+++ b/pandas/io/formats/format.py
@@ -312,8 +312,14 @@ def to_string(self) -> str:
         if len(series) == 0:
             return f"{type(self.series).__name__}([], {footer})"
 
-        have_header = _has_names(series.index)
-        fmt_index = self.tr_series.index.format(name=True)
+        index = series.index
+        have_header = _has_names(index)
+        if isinstance(index, MultiIndex):
+            fmt_index = index._format_multi(include_names=True, sparsify=None)
+            adj = printing.get_adjustment()
+            fmt_index = adj.adjoin(2, *fmt_index).split("\n")
+        else:
+            fmt_index = index._format_flat(include_name=True)
         fmt_values = self._get_formatted_values()
 
         if self.is_truncated_vertically:
@@ -776,7 +782,7 @@ def _get_formatted_column_labels(self, frame: DataFrame) -> list[list[str]]:
         columns = frame.columns
 
         if isinstance(columns, MultiIndex):
-            fmt_columns = columns.format(sparsify=False, adjoin=False)
+            fmt_columns = columns._format_multi(sparsify=False, include_names=False)
             fmt_columns = list(zip(*fmt_columns))
             dtypes = self.frame.dtypes._values
 
@@ -801,7 +807,7 @@ def space_format(x, y):
 
             str_columns = [list(x) for x in zip(*str_columns)]
         else:
-            fmt_columns = columns.format()
+            fmt_columns = columns._format_flat(include_name=False)
             dtypes = self.frame.dtypes
             need_leadsp = dict(zip(fmt_columns, map(is_numeric_dtype, dtypes)))
             str_columns = [
@@ -820,14 +826,15 @@ def _get_formatted_index(self, frame: DataFrame) -> list[str]:
         fmt = self._get_formatter("__index__")
 
         if isinstance(index, MultiIndex):
-            fmt_index = index.format(
+            fmt_index = index._format_multi(
                 sparsify=self.sparsify,
-                adjoin=False,
-                names=self.show_row_idx_names,
+                include_names=self.show_row_idx_names,
                 formatter=fmt,
             )
         else:
-            fmt_index = [index.format(name=self.show_row_idx_names, formatter=fmt)]
+            fmt_index = [
+                index._format_flat(include_name=self.show_row_idx_names, formatter=fmt)
+            ]
 
         fmt_index = [
             tuple(
diff --git a/pandas/io/formats/html.py b/pandas/io/formats/html.py
index b1a3504d46b27..794ce77b3b45e 100644
--- a/pandas/io/formats/html.py
+++ b/pandas/io/formats/html.py
@@ -282,7 +282,7 @@ def _write_col_header(self, indent: int) -> None:
                 sentinel = lib.no_default
             else:
                 sentinel = False
-            levels = self.columns.format(sparsify=sentinel, adjoin=False, names=False)
+            levels = self.columns._format_multi(sparsify=sentinel, include_names=False)
             level_lengths = get_level_lengths(levels, sentinel)
             inner_lvl = len(level_lengths) - 1
             for lnum, (records, values) in enumerate(zip(level_lengths, levels)):
@@ -437,7 +437,8 @@ def _write_regular_rows(
             if fmt is not None:
                 index_values = self.fmt.tr_frame.index.map(fmt)
             else:
-                index_values = self.fmt.tr_frame.index.format()
+                # only reached with non-Multi index
+                index_values = self.fmt.tr_frame.index._format_flat(include_name=False)
 
         row: list[str] = []
         for i in range(nrows):
@@ -480,13 +481,13 @@ def _write_hierarchical_rows(
         nrows = len(frame)
 
         assert isinstance(frame.index, MultiIndex)
-        idx_values = frame.index.format(sparsify=False, adjoin=False, names=False)
+        idx_values = frame.index._format_multi(sparsify=False, include_names=False)
         idx_values = list(zip(*idx_values))
 
         if self.fmt.sparsify:
             # GH3547
             sentinel = lib.no_default
-            levels = frame.index.format(sparsify=sentinel, adjoin=False, names=False)
+            levels = frame.index._format_multi(sparsify=sentinel, include_names=False)
 
             level_lengths = get_level_lengths(levels, sentinel)
             inner_lvl = len(level_lengths) - 1
@@ -579,7 +580,7 @@ def _write_hierarchical_rows(
                     )
 
                 idx_values = list(
-                    zip(*frame.index.format(sparsify=False, adjoin=False, names=False))
+                    zip(*frame.index._format_multi(sparsify=False, include_names=False))
                 )
                 row = []
                 row.extend(idx_values[i])
@@ -606,7 +607,8 @@ def _get_formatted_values(self) -> dict[int, list[str]]:
         return {i: self.fmt.format_col(i) for i in range(self.ncols)}
 
     def _get_columns_formatted_values(self) -> list[str]:
-        return self.columns.format()
+        # only reached with non-Multi Index
+        return self.columns._format_flat(include_name=False)
 
     def write_style(self) -> None:
         # We use the "scoped" attribute here so that the desired
diff --git a/pandas/io/formats/style_render.py b/pandas/io/formats/style_render.py
index 829ed4a33f6a4..416b263ba8497 100644
--- a/pandas/io/formats/style_render.py
+++ b/pandas/io/formats/style_render.py
@@ -1652,9 +1652,9 @@ def _get_level_lengths(
         Result is a dictionary of (level, initial_position): span
     """
     if isinstance(index, MultiIndex):
-        levels = index.format(sparsify=lib.no_default, adjoin=False)
+        levels = index._format_multi(sparsify=lib.no_default, include_names=False)
     else:
-        levels = index.format()
+        levels = index._format_flat(include_name=False)
 
     if hidden_elements is None:
         hidden_elements = []
diff --git a/pandas/tests/indexes/base_class/test_formats.py b/pandas/tests/indexes/base_class/test_formats.py
index 9053d45dee623..20f94010f56f8 100644
--- a/pandas/tests/indexes/base_class/test_formats.py
+++ b/pandas/tests/indexes/base_class/test_formats.py
@@ -4,6 +4,7 @@
 import pandas._config.config as cf
 
 from pandas import Index
+import pandas._testing as tm
 
 
 class TestIndexRendering:
@@ -133,7 +134,9 @@ def test_summary_bug(self):
     def test_index_repr_bool_nan(self):
         # GH32146
         arr = Index([True, False, np.nan], dtype=object)
-        exp1 = arr.format()
+        msg = "Index.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            exp1 = arr.format()
         out1 = ["True", "False", "NaN"]
         assert out1 == exp1
 
@@ -145,4 +148,6 @@ def test_format_different_scalar_lengths(self):
         # GH#35439
         idx = Index(["aaaaaaaaa", "b"])
         expected = ["aaaaaaaaa", "b"]
-        assert idx.format() == expected
+        msg = r"Index\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert idx.format() == expected
diff --git a/pandas/tests/indexes/categorical/test_formats.py b/pandas/tests/indexes/categorical/test_formats.py
index 7dbcaaa8d4ba6..ea3e4ce213e67 100644
--- a/pandas/tests/indexes/categorical/test_formats.py
+++ b/pandas/tests/indexes/categorical/test_formats.py
@@ -4,6 +4,7 @@
 import pandas._config.config as cf
 
 from pandas import CategoricalIndex
+import pandas._testing as tm
 
 
 class TestCategoricalIndexRepr:
@@ -11,7 +12,9 @@ def test_format_different_scalar_lengths(self):
         # GH#35439
         idx = CategoricalIndex(["aaaaaaaaa", "b"])
         expected = ["aaaaaaaaa", "b"]
-        assert idx.format() == expected
+        msg = r"CategoricalIndex\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert idx.format() == expected
 
     def test_string_categorical_index_repr(self):
         # short
diff --git a/pandas/tests/indexes/datetimes/test_datetimelike.py b/pandas/tests/indexes/datetimes/test_datetimelike.py
index a6bee20d3d3ec..a012a2985b41c 100644
--- a/pandas/tests/indexes/datetimes/test_datetimelike.py
+++ b/pandas/tests/indexes/datetimes/test_datetimelike.py
@@ -1,5 +1,6 @@
 """ generic tests from the Datetimelike class """
 from pandas import date_range
+import pandas._testing as tm
 
 
 class TestDatetimeIndex:
@@ -7,4 +8,6 @@ def test_format(self):
         # GH35439
         idx = date_range("20130101", periods=5)
         expected = [f"{x:%Y-%m-%d}" for x in idx]
-        assert idx.format() == expected
+        msg = r"DatetimeIndex\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert idx.format() == expected
diff --git a/pandas/tests/indexes/datetimes/test_formats.py b/pandas/tests/indexes/datetimes/test_formats.py
index 9fb5db9e034ee..6bd978b66c941 100644
--- a/pandas/tests/indexes/datetimes/test_formats.py
+++ b/pandas/tests/indexes/datetimes/test_formats.py
@@ -285,13 +285,17 @@ def test_format_with_name_time_info(self):
         # bug I fixed 12/20/2011
         dates = pd.date_range("2011-01-01 04:00:00", periods=10, name="something")
 
-        formatted = dates.format(name=True)
+        msg = "DatetimeIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = dates.format(name=True)
         assert formatted[0] == "something"
 
     def test_format_datetime_with_time(self):
         dti = DatetimeIndex([datetime(2012, 2, 7), datetime(2012, 2, 7, 23)])
 
-        result = dti.format()
+        msg = "DatetimeIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            result = dti.format()
         expected = ["2012-02-07 00:00:00", "2012-02-07 23:00:00"]
         assert len(result) == 2
         assert result == expected
diff --git a/pandas/tests/indexes/multi/test_formats.py b/pandas/tests/indexes/multi/test_formats.py
index 011f61fac90e8..bbe94824eefa1 100644
--- a/pandas/tests/indexes/multi/test_formats.py
+++ b/pandas/tests/indexes/multi/test_formats.py
@@ -6,24 +6,31 @@
     Index,
     MultiIndex,
 )
+import pandas._testing as tm
 
 
 def test_format(idx):
-    idx.format()
-    idx[:0].format()
+    msg = "MultiIndex.format is deprecated"
+    with tm.assert_produces_warning(FutureWarning, match=msg):
+        idx.format()
+        idx[:0].format()
 
 
 def test_format_integer_names():
     index = MultiIndex(
         levels=[[0, 1], [0, 1]], codes=[[0, 0, 1, 1], [0, 1, 0, 1]], names=[0, 1]
     )
-    index.format(names=True)
+    msg = "MultiIndex.format is deprecated"
+    with tm.assert_produces_warning(FutureWarning, match=msg):
+        index.format(names=True)
 
 
 def test_format_sparse_config(idx):
     # GH1538
+    msg = "MultiIndex.format is deprecated"
     with pd.option_context("display.multi_sparse", False):
-        result = idx.format()
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            result = idx.format()
     assert result[1] == "foo  two"
 
 
@@ -37,8 +44,9 @@ def test_format_sparse_display():
             [0, 0, 0, 0, 0, 0],
         ],
     )
-
-    result = index.format()
+    msg = "MultiIndex.format is deprecated"
+    with tm.assert_produces_warning(FutureWarning, match=msg):
+        result = index.format()
     assert result[3] == "1  0  0  0"
 
 
diff --git a/pandas/tests/indexes/period/test_period.py b/pandas/tests/indexes/period/test_period.py
index 22bb63d67f57f..1bb8d66332cd0 100644
--- a/pandas/tests/indexes/period/test_period.py
+++ b/pandas/tests/indexes/period/test_period.py
@@ -275,8 +275,11 @@ def test_map(self):
     def test_format_empty(self):
         # GH35712
         empty_idx = PeriodIndex([], freq="Y")
-        assert empty_idx.format() == []
-        assert empty_idx.format(name=True) == [""]
+        msg = r"PeriodIndex\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert empty_idx.format() == []
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert empty_idx.format(name=True) == [""]
 
     def test_period_index_frequency_ME_error_message(self):
         msg = "Invalid frequency: 2ME"
diff --git a/pandas/tests/indexes/ranges/test_range.py b/pandas/tests/indexes/ranges/test_range.py
index 132704434829e..95756b04bca69 100644
--- a/pandas/tests/indexes/ranges/test_range.py
+++ b/pandas/tests/indexes/ranges/test_range.py
@@ -240,7 +240,9 @@ def test_cache(self):
             pass
         assert idx._cache == {}
 
-        idx.format()
+        msg = "RangeIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            idx.format()
         assert idx._cache == {}
 
         df = pd.DataFrame({"a": range(10)}, index=idx)
@@ -566,8 +568,11 @@ def test_engineless_lookup(self):
     def test_format_empty(self):
         # GH35712
         empty_idx = RangeIndex(0)
-        assert empty_idx.format() == []
-        assert empty_idx.format(name=True) == [""]
+        msg = r"RangeIndex\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert empty_idx.format() == []
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert empty_idx.format(name=True) == [""]
 
     @pytest.mark.parametrize(
         "ri",
diff --git a/pandas/tests/indexes/test_base.py b/pandas/tests/indexes/test_base.py
index 6afab569797f2..04ab2020b4c7a 100644
--- a/pandas/tests/indexes/test_base.py
+++ b/pandas/tests/indexes/test_base.py
@@ -666,13 +666,17 @@ def test_format_bug(self):
         # include us since the default for Timestamp shows these but Index
         # formatting does not we are skipping)
         now = datetime.now()
+        msg = r"Index\.format is deprecated"
+
         if not str(now).endswith("000"):
             index = Index([now])
-            formatted = index.format()
+            with tm.assert_produces_warning(FutureWarning, match=msg):
+                formatted = index.format()
             expected = [str(index[0])]
             assert formatted == expected
 
-        Index([]).format()
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            Index([]).format()
 
     @pytest.mark.parametrize("vals", [[1, 2.0 + 3.0j, 4.0], ["a", "b", "c"]])
     def test_format_missing(self, vals, nulls_fixture):
@@ -682,7 +686,9 @@ def test_format_missing(self, vals, nulls_fixture):
         index = Index(vals, dtype=object)
         # TODO: case with complex dtype?
 
-        formatted = index.format()
+        msg = r"Index\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = index.format()
         null_repr = "NaN" if isinstance(nulls_fixture, float) else str(nulls_fixture)
         expected = [str(index[0]), str(index[1]), str(index[2]), null_repr]
 
diff --git a/pandas/tests/indexes/test_old_base.py b/pandas/tests/indexes/test_old_base.py
index 79dc423f12a85..32adbc693390b 100644
--- a/pandas/tests/indexes/test_old_base.py
+++ b/pandas/tests/indexes/test_old_base.py
@@ -559,15 +559,20 @@ def test_format(self, simple_index):
             pytest.skip("Tested elsewhere.")
         idx = simple_index
         expected = [str(x) for x in idx]
-        assert idx.format() == expected
+        msg = r"Index\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert idx.format() == expected
 
     def test_format_empty(self, simple_index):
         # GH35712
         if isinstance(simple_index, (PeriodIndex, RangeIndex)):
             pytest.skip("Tested elsewhere")
         empty_idx = type(simple_index)([])
-        assert empty_idx.format() == []
-        assert empty_idx.format(name=True) == [""]
+        msg = r"Index\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert empty_idx.format() == []
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert empty_idx.format(name=True) == [""]
 
     def test_fillna(self, index):
         # GH 11343
@@ -955,7 +960,9 @@ def test_format(self, simple_index):
         idx = simple_index
         max_width = max(len(str(x)) for x in idx)
         expected = [str(x).ljust(max_width) for x in idx]
-        assert idx.format() == expected
+        msg = r"Index\.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert idx.format() == expected
 
     def test_insert_non_na(self, simple_index):
         # GH#43921 inserting an element that we know we can hold should
diff --git a/pandas/tests/io/excel/test_writers.py b/pandas/tests/io/excel/test_writers.py
index e4ce969daab53..18af18ade85f4 100644
--- a/pandas/tests/io/excel/test_writers.py
+++ b/pandas/tests/io/excel/test_writers.py
@@ -809,7 +809,7 @@ def test_to_excel_multiindex_cols(self, merge_cells, frame, path):
                 reader, sheet_name="test1", header=header, index_col=[0, 1]
             )
         if not merge_cells:
-            fm = frame.columns.format(sparsify=False, adjoin=False, names=False)
+            fm = frame.columns._format_multi(sparsify=False, include_names=False)
             frame.columns = [".".join(map(str, q)) for q in zip(*fm)]
         tm.assert_frame_equal(frame, df)
 
diff --git a/pandas/tests/io/formats/test_format.py b/pandas/tests/io/formats/test_format.py
index 57f1f082708ae..d18c333d79244 100644
--- a/pandas/tests/io/formats/test_format.py
+++ b/pandas/tests/io/formats/test_format.py
@@ -3333,7 +3333,9 @@ def test_period_format_and_strftime_default(self):
         per = pd.PeriodIndex([datetime(2003, 1, 1, 12), None], freq="h")
 
         # Default formatting
-        formatted = per.format()
+        msg = "PeriodIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = per.format()
         assert formatted[0] == "2003-01-01 12:00"  # default: minutes not shown
         assert formatted[1] == "NaT"
         # format is equivalent to strftime(None)...
@@ -3342,35 +3344,40 @@ def test_period_format_and_strftime_default(self):
 
         # Same test with nanoseconds freq
         per = pd.period_range("2003-01-01 12:01:01.123456789", periods=2, freq="ns")
-        formatted = per.format()
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = per.format()
         assert (formatted == per.strftime(None)).all()
         assert formatted[0] == "2003-01-01 12:01:01.123456789"
         assert formatted[1] == "2003-01-01 12:01:01.123456790"
 
     def test_period_custom(self):
         # GH#46252 custom formatting directives %l (ms) and %u (us)
+        msg = "PeriodIndex.format is deprecated"
 
         # 3 digits
         per = pd.period_range("2003-01-01 12:01:01.123", periods=2, freq="ms")
-        formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
         assert formatted[0] == "03 12:01:01 (ms=123 us=123000 ns=123000000)"
         assert formatted[1] == "03 12:01:01 (ms=124 us=124000 ns=124000000)"
 
         # 6 digits
         per = pd.period_range("2003-01-01 12:01:01.123456", periods=2, freq="us")
-        formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
         assert formatted[0] == "03 12:01:01 (ms=123 us=123456 ns=123456000)"
         assert formatted[1] == "03 12:01:01 (ms=123 us=123457 ns=123457000)"
 
         # 9 digits
         per = pd.period_range("2003-01-01 12:01:01.123456789", periods=2, freq="ns")
-        formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
         assert formatted[0] == "03 12:01:01 (ms=123 us=123456 ns=123456789)"
         assert formatted[1] == "03 12:01:01 (ms=123 us=123456 ns=123456790)"
 
     def test_period_tz(self):
         # Formatting periods created from a datetime with timezone.
-
+        msg = r"PeriodIndex\.format is deprecated"
         # This timestamp is in 2013 in Europe/Paris but is 2012 in UTC
         dt = pd.to_datetime(["2013-01-01 00:00:00+01:00"], utc=True)
 
@@ -3378,13 +3385,15 @@ def test_period_tz(self):
         # Since tz is currently set as utc, we'll see 2012
         with tm.assert_produces_warning(UserWarning, match="will drop timezone"):
             per = dt.to_period(freq="h")
-        assert per.format()[0] == "2012-12-31 23:00"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert per.format()[0] == "2012-12-31 23:00"
 
         # If tz is currently set as paris before conversion, we'll see 2013
         dt = dt.tz_convert("Europe/Paris")
         with tm.assert_produces_warning(UserWarning, match="will drop timezone"):
             per = dt.to_period(freq="h")
-        assert per.format()[0] == "2013-01-01 00:00"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            assert per.format()[0] == "2013-01-01 00:00"
 
     @pytest.mark.parametrize(
         "locale_str",
@@ -3411,7 +3420,9 @@ def test_period_non_ascii_fmt(self, locale_str):
 
             # Index
             per = pd.period_range("2003-01-01 01:00:00", periods=2, freq="12h")
-            formatted = per.format(date_format="%y é")
+            msg = "PeriodIndex.format is deprecated"
+            with tm.assert_produces_warning(FutureWarning, match=msg):
+                formatted = per.format(date_format="%y é")
             assert formatted[0] == "03 é"
             assert formatted[1] == "03 é"
 
@@ -3443,33 +3454,45 @@ def test_period_custom_locale_directive(self, locale_str):
 
             # Index
             per = pd.period_range("2003-01-01 01:00:00", periods=2, freq="12h")
-            formatted = per.format(date_format="%y %I:%M:%S%p")
+            msg = "PeriodIndex.format is deprecated"
+            with tm.assert_produces_warning(FutureWarning, match=msg):
+                formatted = per.format(date_format="%y %I:%M:%S%p")
             assert formatted[0] == f"03 01:00:00{am_local}"
             assert formatted[1] == f"03 01:00:00{pm_local}"
 
 
 class TestDatetimeIndexFormat:
     def test_datetime(self):
-        formatted = pd.to_datetime([datetime(2003, 1, 1, 12), NaT]).format()
+        msg = "DatetimeIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = pd.to_datetime([datetime(2003, 1, 1, 12), NaT]).format()
         assert formatted[0] == "2003-01-01 12:00:00"
         assert formatted[1] == "NaT"
 
     def test_date(self):
-        formatted = pd.to_datetime([datetime(2003, 1, 1), NaT]).format()
+        msg = "DatetimeIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = pd.to_datetime([datetime(2003, 1, 1), NaT]).format()
         assert formatted[0] == "2003-01-01"
         assert formatted[1] == "NaT"
 
     def test_date_tz(self):
-        formatted = pd.to_datetime([datetime(2013, 1, 1)], utc=True).format()
+        dti = pd.to_datetime([datetime(2013, 1, 1)], utc=True)
+        msg = "DatetimeIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = dti.format()
         assert formatted[0] == "2013-01-01 00:00:00+00:00"
 
-        formatted = pd.to_datetime([datetime(2013, 1, 1), NaT], utc=True).format()
+        dti = pd.to_datetime([datetime(2013, 1, 1), NaT], utc=True)
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = dti.format()
         assert formatted[0] == "2013-01-01 00:00:00+00:00"
 
     def test_date_explicit_date_format(self):
-        formatted = pd.to_datetime([datetime(2003, 2, 1), NaT]).format(
-            date_format="%m-%d-%Y", na_rep="UT"
-        )
+        dti = pd.to_datetime([datetime(2003, 2, 1), NaT])
+        msg = "DatetimeIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            formatted = dti.format(date_format="%m-%d-%Y", na_rep="UT")
         assert formatted[0] == "02-01-2003"
         assert formatted[1] == "UT"
 
diff --git a/pandas/tests/series/test_repr.py b/pandas/tests/series/test_repr.py
index 86474a38d29fb..86addb9dadfad 100644
--- a/pandas/tests/series/test_repr.py
+++ b/pandas/tests/series/test_repr.py
@@ -258,7 +258,9 @@ def test_index_repr_in_frame_with_nan(self):
 
     def test_format_pre_1900_dates(self):
         rng = date_range("1/1/1850", "1/1/1950", freq="Y-DEC")
-        rng.format()
+        msg = "DatetimeIndex.format is deprecated"
+        with tm.assert_produces_warning(FutureWarning, match=msg):
+            rng.format()
         ts = Series(1, index=rng)
         repr(ts)