diff --git a/doc/source/whatsnew/v2.0.0.rst b/doc/source/whatsnew/v2.0.0.rst
index bea7dd0a1ed6d..6bc0cb2969734 100644
--- a/doc/source/whatsnew/v2.0.0.rst
+++ b/doc/source/whatsnew/v2.0.0.rst
@@ -223,6 +223,9 @@ Copy-on-Write improvements
   - :meth:`DataFrame.to_period` / :meth:`Series.to_period`
   - :meth:`DataFrame.truncate`
   - :meth:`DataFrame.tz_convert` / :meth:`Series.tz_localize`
+  - :meth:`DataFrame.interpolate` / :meth:`Series.interpolate`
+  - :meth:`DataFrame.ffill` / :meth:`Series.ffill`
+  - :meth:`DataFrame.bfill` / :meth:`Series.bfill`
   - :meth:`DataFrame.infer_objects` / :meth:`Series.infer_objects`
   - :meth:`DataFrame.astype` / :meth:`Series.astype`
   - :func:`concat`
diff --git a/pandas/core/internals/blocks.py b/pandas/core/internals/blocks.py
index e66011acb978b..5ccc0aeb03500 100644
--- a/pandas/core/internals/blocks.py
+++ b/pandas/core/internals/blocks.py
@@ -228,7 +228,10 @@ def make_block(
 
     @final
     def make_block_same_class(
-        self, values, placement: BlockPlacement | None = None
+        self,
+        values,
+        placement: BlockPlacement | None = None,
+        refs: BlockValuesRefs | None = None,
     ) -> Block:
         """Wrap given values in a block of same type as self."""
         # Pre-2.0 we called ensure_wrapped_if_datetimelike because fastparquet
@@ -237,7 +240,7 @@ def make_block_same_class(
             placement = self._mgr_locs
 
         # We assume maybe_coerce_values has already been called
-        return type(self)(values, placement=placement, ndim=self.ndim)
+        return type(self)(values, placement=placement, ndim=self.ndim, refs=refs)
 
     @final
     def __repr__(self) -> str:
@@ -421,7 +424,9 @@ def coerce_to_target_dtype(self, other) -> Block:
         return self.astype(new_dtype, copy=False)
 
     @final
-    def _maybe_downcast(self, blocks: list[Block], downcast=None) -> list[Block]:
+    def _maybe_downcast(
+        self, blocks: list[Block], downcast=None, using_cow: bool = False
+    ) -> list[Block]:
         if downcast is False:
             return blocks
 
@@ -431,23 +436,26 @@ def _maybe_downcast(self, blocks: list[Block], downcast=None) -> list[Block]:
             #  but ATM it breaks too much existing code.
             # split and convert the blocks
 
-            return extend_blocks([blk.convert() for blk in blocks])
+            return extend_blocks(
+                [blk.convert(using_cow=using_cow, copy=not using_cow) for blk in blocks]
+            )
 
         if downcast is None:
             return blocks
 
-        return extend_blocks([b._downcast_2d(downcast) for b in blocks])
+        return extend_blocks([b._downcast_2d(downcast, using_cow) for b in blocks])
 
     @final
     @maybe_split
-    def _downcast_2d(self, dtype) -> list[Block]:
+    def _downcast_2d(self, dtype, using_cow: bool = False) -> list[Block]:
         """
         downcast specialized to 2D case post-validation.
 
         Refactored to allow use of maybe_split.
         """
         new_values = maybe_downcast_to_dtype(self.values, dtype=dtype)
-        return [self.make_block(new_values)]
+        refs = self.refs if using_cow and new_values is self.values else None
+        return [self.make_block(new_values, refs=refs)]
 
     def convert(
         self,
@@ -1209,6 +1217,7 @@ def interpolate(
         limit_area: str | None = None,
         fill_value: Any | None = None,
         downcast: str | None = None,
+        using_cow: bool = False,
         **kwargs,
     ) -> list[Block]:
 
@@ -1216,6 +1225,8 @@ def interpolate(
 
         if not self._can_hold_na:
             # If there are no NAs, then interpolate is a no-op
+            if using_cow:
+                return [self.copy(deep=False)]
             return [self] if inplace else [self.copy()]
 
         try:
@@ -1224,8 +1235,10 @@ def interpolate(
             m = None
         if m is None and self.dtype.kind != "f":
             # only deal with floats
-            # bc we already checked that can_hold_na, we dont have int dtype here
+            # bc we already checked that can_hold_na, we don't have int dtype here
             # test_interp_basic checks that we make a copy here
+            if using_cow:
+                return [self.copy(deep=False)]
             return [self] if inplace else [self.copy()]
 
         if self.is_object and self.ndim == 2 and self.shape[0] != 1 and axis == 0:
@@ -1244,7 +1257,15 @@ def interpolate(
                 **kwargs,
             )
 
-        data = self.values if inplace else self.values.copy()
+        refs = None
+        if inplace:
+            if using_cow and self.refs.has_reference():
+                data = self.values.copy()
+            else:
+                data = self.values
+                refs = self.refs
+        else:
+            data = self.values.copy()
         data = cast(np.ndarray, data)  # bc overridden by ExtensionBlock
 
         missing.interpolate_array_2d(
@@ -1259,8 +1280,8 @@ def interpolate(
             **kwargs,
         )
 
-        nb = self.make_block_same_class(data)
-        return nb._maybe_downcast([nb], downcast)
+        nb = self.make_block_same_class(data, refs=refs)
+        return nb._maybe_downcast([nb], downcast, using_cow)
 
     def diff(self, n: int, axis: AxisInt = 1) -> list[Block]:
         """return block for the diff of the values"""
@@ -1632,6 +1653,7 @@ def interpolate(
         inplace: bool = False,
         limit: int | None = None,
         fill_value=None,
+        using_cow: bool = False,
         **kwargs,
     ):
         values = self.values
@@ -2011,6 +2033,7 @@ def interpolate(
         inplace: bool = False,
         limit: int | None = None,
         fill_value=None,
+        using_cow: bool = False,
         **kwargs,
     ):
         values = self.values
@@ -2020,12 +2043,20 @@ def interpolate(
         # "Literal['linear']")  [comparison-overlap]
         if method == "linear":  # type: ignore[comparison-overlap]
             # TODO: GH#50950 implement for arbitrary EAs
-            data_out = values._ndarray if inplace else values._ndarray.copy()
+            refs = None
+            if using_cow:
+                if inplace and not self.refs.has_reference():
+                    data_out = values._ndarray
+                    refs = self.refs
+                else:
+                    data_out = values._ndarray.copy()
+            else:
+                data_out = values._ndarray if inplace else values._ndarray.copy()
             missing.interpolate_array_2d(
                 data_out, method=method, limit=limit, index=index, axis=axis
             )
             new_values = type(values)._simple_new(data_out, dtype=values.dtype)
-            return self.make_block_same_class(new_values)
+            return self.make_block_same_class(new_values, refs=refs)
 
         elif values.ndim == 2 and axis == 0:
             # NDArrayBackedExtensionArray.fillna assumes axis=1
diff --git a/pandas/core/internals/managers.py b/pandas/core/internals/managers.py
index 517e6d7e48275..3601337b4cb9c 100644
--- a/pandas/core/internals/managers.py
+++ b/pandas/core/internals/managers.py
@@ -389,14 +389,9 @@ def diff(self: T, n: int, axis: AxisInt) -> T:
         return self.apply("diff", n=n, axis=axis)
 
     def interpolate(self: T, inplace: bool, **kwargs) -> T:
-        if inplace:
-            # TODO(CoW) can be optimized to only copy those blocks that have refs
-            if using_copy_on_write() and any(
-                not self._has_no_reference_block(i) for i in range(len(self.blocks))
-            ):
-                self = self.copy()
-
-        return self.apply("interpolate", inplace=inplace, **kwargs)
+        return self.apply(
+            "interpolate", inplace=inplace, **kwargs, using_cow=using_copy_on_write()
+        )
 
     def shift(self: T, periods: int, axis: AxisInt, fill_value) -> T:
         axis = self._normalize_axis(axis)
diff --git a/pandas/tests/copy_view/test_interp_fillna.py b/pandas/tests/copy_view/test_interp_fillna.py
new file mode 100644
index 0000000000000..1bfcf7d180e54
--- /dev/null
+++ b/pandas/tests/copy_view/test_interp_fillna.py
@@ -0,0 +1,164 @@
+import numpy as np
+import pytest
+
+from pandas import (
+    DataFrame,
+    NaT,
+    Series,
+    Timestamp,
+)
+import pandas._testing as tm
+from pandas.tests.copy_view.util import get_array
+
+
+@pytest.mark.parametrize("method", ["pad", "nearest", "linear"])
+def test_interpolate_no_op(using_copy_on_write, method):
+    df = DataFrame({"a": [1, 2]})
+    df_orig = df.copy()
+
+    result = df.interpolate(method=method)
+
+    if using_copy_on_write:
+        assert np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+    else:
+        assert not np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+
+    result.iloc[0, 0] = 100
+
+    if using_copy_on_write:
+        assert not np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+    tm.assert_frame_equal(df, df_orig)
+
+
+@pytest.mark.parametrize("func", ["ffill", "bfill"])
+def test_interp_fill_functions(using_copy_on_write, func):
+    # Check that these takes the same code paths as interpolate
+    df = DataFrame({"a": [1, 2]})
+    df_orig = df.copy()
+
+    result = getattr(df, func)()
+
+    if using_copy_on_write:
+        assert np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+    else:
+        assert not np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+
+    result.iloc[0, 0] = 100
+
+    if using_copy_on_write:
+        assert not np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+    tm.assert_frame_equal(df, df_orig)
+
+
+@pytest.mark.parametrize("func", ["ffill", "bfill"])
+@pytest.mark.parametrize(
+    "vals", [[1, np.nan, 2], [Timestamp("2019-12-31"), NaT, Timestamp("2020-12-31")]]
+)
+def test_interpolate_triggers_copy(using_copy_on_write, vals, func):
+    df = DataFrame({"a": vals})
+    result = getattr(df, func)()
+
+    assert not np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+    if using_copy_on_write:
+        # Check that we don't have references when triggering a copy
+        assert result._mgr._has_no_reference(0)
+
+
+@pytest.mark.parametrize(
+    "vals", [[1, np.nan, 2], [Timestamp("2019-12-31"), NaT, Timestamp("2020-12-31")]]
+)
+def test_interpolate_inplace_no_reference_no_copy(using_copy_on_write, vals):
+    df = DataFrame({"a": vals})
+    arr = get_array(df, "a")
+    df.interpolate(method="linear", inplace=True)
+
+    assert np.shares_memory(arr, get_array(df, "a"))
+    if using_copy_on_write:
+        # Check that we don't have references when triggering a copy
+        assert df._mgr._has_no_reference(0)
+
+
+@pytest.mark.parametrize(
+    "vals", [[1, np.nan, 2], [Timestamp("2019-12-31"), NaT, Timestamp("2020-12-31")]]
+)
+def test_interpolate_inplace_with_refs(using_copy_on_write, vals):
+    df = DataFrame({"a": [1, np.nan, 2]})
+    df_orig = df.copy()
+    arr = get_array(df, "a")
+    view = df[:]
+    df.interpolate(method="linear", inplace=True)
+
+    if using_copy_on_write:
+        # Check that copy was triggered in interpolate and that we don't
+        # have any references left
+        assert not np.shares_memory(arr, get_array(df, "a"))
+        tm.assert_frame_equal(df_orig, view)
+        assert df._mgr._has_no_reference(0)
+        assert view._mgr._has_no_reference(0)
+    else:
+        assert np.shares_memory(arr, get_array(df, "a"))
+
+
+def test_interpolate_cleaned_fill_method(using_copy_on_write):
+    # Check that "method is set to None" case works correctly
+    df = DataFrame({"a": ["a", np.nan, "c"], "b": 1})
+    df_orig = df.copy()
+
+    result = df.interpolate(method="asfreq")
+
+    if using_copy_on_write:
+        assert np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+    else:
+        assert not np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+
+    result.iloc[0, 0] = Timestamp("2021-12-31")
+
+    if using_copy_on_write:
+        assert not np.shares_memory(get_array(result, "a"), get_array(df, "a"))
+    tm.assert_frame_equal(df, df_orig)
+
+
+def test_interpolate_object_convert_no_op(using_copy_on_write):
+    df = DataFrame({"a": ["a", "b", "c"], "b": 1})
+    arr_a = get_array(df, "a")
+    df.interpolate(method="pad", inplace=True)
+
+    # Now CoW makes a copy, it should not!
+    if using_copy_on_write:
+        assert df._mgr._has_no_reference(0)
+        assert np.shares_memory(arr_a, get_array(df, "a"))
+
+
+def test_interpolate_object_convert_copies(using_copy_on_write):
+    df = DataFrame({"a": Series([1, 2], dtype=object), "b": 1})
+    arr_a = get_array(df, "a")
+    df.interpolate(method="pad", inplace=True)
+
+    if using_copy_on_write:
+        assert df._mgr._has_no_reference(0)
+        assert not np.shares_memory(arr_a, get_array(df, "a"))
+
+
+def test_interpolate_downcast(using_copy_on_write):
+    df = DataFrame({"a": [1, np.nan, 2.5], "b": 1})
+    arr_a = get_array(df, "a")
+    df.interpolate(method="pad", inplace=True, downcast="infer")
+
+    if using_copy_on_write:
+        assert df._mgr._has_no_reference(0)
+    assert np.shares_memory(arr_a, get_array(df, "a"))
+
+
+def test_interpolate_downcast_reference_triggers_copy(using_copy_on_write):
+    df = DataFrame({"a": [1, np.nan, 2.5], "b": 1})
+    df_orig = df.copy()
+    arr_a = get_array(df, "a")
+    view = df[:]
+    df.interpolate(method="pad", inplace=True, downcast="infer")
+
+    if using_copy_on_write:
+        assert df._mgr._has_no_reference(0)
+        assert not np.shares_memory(arr_a, get_array(df, "a"))
+        tm.assert_frame_equal(df_orig, view)
+    else:
+        tm.assert_frame_equal(df, view)
diff --git a/pandas/tests/frame/methods/test_interpolate.py b/pandas/tests/frame/methods/test_interpolate.py
index 20e4571e2ba05..d6da967106fe6 100644
--- a/pandas/tests/frame/methods/test_interpolate.py
+++ b/pandas/tests/frame/methods/test_interpolate.py
@@ -52,7 +52,7 @@ def test_interpolate_inplace(self, frame_or_series, using_array_manager, request
         assert np.shares_memory(orig, obj.values)
         assert orig.squeeze()[1] == 1.5
 
-    def test_interp_basic(self):
+    def test_interp_basic(self, using_copy_on_write):
         df = DataFrame(
             {
                 "A": [1, 2, np.nan, 4],
@@ -75,8 +75,12 @@ def test_interp_basic(self):
         # check we didn't operate inplace GH#45791
         cvalues = df["C"]._values
         dvalues = df["D"].values
-        assert not np.shares_memory(cvalues, result["C"]._values)
-        assert not np.shares_memory(dvalues, result["D"]._values)
+        if using_copy_on_write:
+            assert np.shares_memory(cvalues, result["C"]._values)
+            assert np.shares_memory(dvalues, result["D"]._values)
+        else:
+            assert not np.shares_memory(cvalues, result["C"]._values)
+            assert not np.shares_memory(dvalues, result["D"]._values)
 
         res = df.interpolate(inplace=True)
         assert res is None
@@ -95,14 +99,6 @@ def test_interp_basic_with_non_range_index(self):
                 "D": list("abcd"),
             }
         )
-        expected = DataFrame(
-            {
-                "A": [1.0, 2.0, 3.0, 4.0],
-                "B": [1.0, 4.0, 9.0, 9.0],
-                "C": [1, 2, 3, 5],
-                "D": list("abcd"),
-            }
-        )
 
         result = df.set_index("C").interpolate()
         expected = df.set_index("C")
@@ -327,20 +323,24 @@ def test_interp_raise_on_all_object_dtype(self):
             df.interpolate()
 
     def test_interp_inplace(self, using_copy_on_write):
-        # TODO(CoW) inplace keyword (it is still mutating the parent)
-        if using_copy_on_write:
-            pytest.skip("CoW: inplace keyword not yet handled")
         df = DataFrame({"a": [1.0, 2.0, np.nan, 4.0]})
         expected = DataFrame({"a": [1.0, 2.0, 3.0, 4.0]})
+        expected_cow = df.copy()
         result = df.copy()
         return_value = result["a"].interpolate(inplace=True)
         assert return_value is None
-        tm.assert_frame_equal(result, expected)
+        if using_copy_on_write:
+            tm.assert_frame_equal(result, expected_cow)
+        else:
+            tm.assert_frame_equal(result, expected)
 
         result = df.copy()
         return_value = result["a"].interpolate(inplace=True, downcast="infer")
         assert return_value is None
-        tm.assert_frame_equal(result, expected.astype("int64"))
+        if using_copy_on_write:
+            tm.assert_frame_equal(result, expected_cow)
+        else:
+            tm.assert_frame_equal(result, expected.astype("int64"))
 
     def test_interp_inplace_row(self):
         # GH 10395