diff --git a/doc/source/whatsnew/v1.0.1.rst b/doc/source/whatsnew/v1.0.1.rst
index f9c756b2518af..20cfcfbde389c 100644
--- a/doc/source/whatsnew/v1.0.1.rst
+++ b/doc/source/whatsnew/v1.0.1.rst
@@ -20,6 +20,7 @@ Fixed regressions
 - Fixed regression in ``DataFrame.__setitem__`` raising an ``AttributeError`` with a :class:`MultiIndex` and a non-monotonic indexer (:issue:`31449`)
 - Fixed regression in :class:`Series` multiplication when multiplying a numeric :class:`Series` with >10000 elements with a timedelta-like scalar (:issue:`31457`)
 - Fixed regression in :meth:`GroupBy.apply` if called with a function which returned a non-pandas non-scalar object (e.g. a list or numpy array) (:issue:`31441`)
+- Fixed regression in :meth:`DataFrame.groupby` whereby taking the minimum or maximum of a column with period dtype would raise a ``TypeError``. (:issue:`31471`)
 - Fixed regression in :meth:`to_datetime` when parsing non-nanosecond resolution datetimes (:issue:`31491`)
 - Fixed regression in :meth:`~DataFrame.to_csv` where specifying an ``na_rep`` might truncate the values written (:issue:`31447`)
 - Fixed regression in :class:`Categorical` construction with ``numpy.str_`` categories (:issue:`31499`)
diff --git a/pandas/core/groupby/ops.py b/pandas/core/groupby/ops.py
index 77c54ec736aaa..761353ca5a6ca 100644
--- a/pandas/core/groupby/ops.py
+++ b/pandas/core/groupby/ops.py
@@ -31,6 +31,7 @@
     is_extension_array_dtype,
     is_integer_dtype,
     is_numeric_dtype,
+    is_period_dtype,
     is_sparse,
     is_timedelta64_dtype,
     needs_i8_conversion,
@@ -567,7 +568,12 @@ def _cython_operation(
         if swapped:
             result = result.swapaxes(0, axis)
 
-        if is_datetime64tz_dtype(orig_values.dtype):
+        if is_datetime64tz_dtype(orig_values.dtype) or is_period_dtype(
+            orig_values.dtype
+        ):
+            # We need to use the constructors directly for these dtypes
+            # since numpy won't recognize them
+            # https://github.com/pandas-dev/pandas/issues/31471
             result = type(orig_values)(result.astype(np.int64), dtype=orig_values.dtype)
         elif is_datetimelike and kind == "aggregate":
             result = result.astype(orig_values.dtype)
diff --git a/pandas/tests/groupby/aggregate/test_aggregate.py b/pandas/tests/groupby/aggregate/test_aggregate.py
index 2d31996a8a964..71af6533db764 100644
--- a/pandas/tests/groupby/aggregate/test_aggregate.py
+++ b/pandas/tests/groupby/aggregate/test_aggregate.py
@@ -684,6 +684,34 @@ def aggfunc(x):
     tm.assert_frame_equal(result, expected)
 
 
+@pytest.mark.parametrize("func", ["min", "max"])
+def test_groupby_aggregate_period_column(func):
+    # GH 31471
+    groups = [1, 2]
+    periods = pd.period_range("2020", periods=2, freq="Y")
+    df = pd.DataFrame({"a": groups, "b": periods})
+
+    result = getattr(df.groupby("a")["b"], func)()
+    idx = pd.Int64Index([1, 2], name="a")
+    expected = pd.Series(periods, index=idx, name="b")
+
+    tm.assert_series_equal(result, expected)
+
+
+@pytest.mark.parametrize("func", ["min", "max"])
+def test_groupby_aggregate_period_frame(func):
+    # GH 31471
+    groups = [1, 2]
+    periods = pd.period_range("2020", periods=2, freq="Y")
+    df = pd.DataFrame({"a": groups, "b": periods})
+
+    result = getattr(df.groupby("a"), func)()
+    idx = pd.Int64Index([1, 2], name="a")
+    expected = pd.DataFrame({"b": periods}, index=idx)
+
+    tm.assert_frame_equal(result, expected)
+
+
 class TestLambdaMangling:
     def test_basic(self):
         df = pd.DataFrame({"A": [0, 0, 1, 1], "B": [1, 2, 3, 4]})