diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index 812af544ed9d8..83614d7a9628b 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -225,6 +225,7 @@ Other enhancements
 - :class:`DataFrame` now supports ``divmod`` operation (:issue:`37165`)
 - :meth:`DataFrame.to_parquet` now returns a ``bytes`` object when no ``path`` argument is passed (:issue:`37105`)
 - :class:`Rolling` now supports the ``closed`` argument for fixed windows (:issue:`34315`)
+- :class:`DatetimeIndex` and :class:`Series` with ``datetime64`` or ``datetime64tz`` dtypes now support ``std`` (:issue:`37436`)
 
 .. _whatsnew_120.api_breaking.python:
 
diff --git a/pandas/core/arrays/datetimes.py b/pandas/core/arrays/datetimes.py
index 751290d5af9ae..a1050f4271e05 100644
--- a/pandas/core/arrays/datetimes.py
+++ b/pandas/core/arrays/datetimes.py
@@ -1862,6 +1862,28 @@ def to_julian_date(self):
             / 24.0
         )
 
+    # -----------------------------------------------------------------
+    # Reductions
+
+    def std(
+        self,
+        axis=None,
+        dtype=None,
+        out=None,
+        ddof: int = 1,
+        keepdims: bool = False,
+        skipna: bool = True,
+    ):
+        # Because std is translation-invariant, we can get self.std
+        #  by calculating (self - Timestamp(0)).std, and we can do it
+        #  without creating a copy by using a view on self._ndarray
+        from pandas.core.arrays import TimedeltaArray
+
+        tda = TimedeltaArray(self._ndarray.view("i8"))
+        return tda.std(
+            axis=axis, dtype=dtype, out=out, ddof=ddof, keepdims=keepdims, skipna=skipna
+        )
+
 
 # -------------------------------------------------------------------
 # Constructor Helpers
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index 005272750997e..aa16dc9752565 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -99,6 +99,7 @@ def _new_DatetimeIndex(cls, d):
         "date",
         "time",
         "timetz",
+        "std",
     ]
     + DatetimeArray._bool_ops,
     DatetimeArray,
@@ -201,6 +202,7 @@ class DatetimeIndex(DatetimeTimedeltaMixin):
     month_name
     day_name
     mean
+    std
 
     See Also
     --------
diff --git a/pandas/core/nanops.py b/pandas/core/nanops.py
index 3ab08ab2cda56..46ff4a0e2f612 100644
--- a/pandas/core/nanops.py
+++ b/pandas/core/nanops.py
@@ -756,7 +756,6 @@ def _get_counts_nanvar(
     return count, d
 
 
-@disallow("M8")
 @bottleneck_switch(ddof=1)
 def nanstd(values, axis=None, skipna=True, ddof=1, mask=None):
     """
@@ -786,6 +785,9 @@ def nanstd(values, axis=None, skipna=True, ddof=1, mask=None):
     >>> nanops.nanstd(s)
     1.0
     """
+    if values.dtype == "M8[ns]":
+        values = values.view("m8[ns]")
+
     orig_dtype = values.dtype
     values, mask, _, _, _ = _get_values(values, skipna, mask=mask)
 
diff --git a/pandas/tests/arrays/test_timedeltas.py b/pandas/tests/arrays/test_timedeltas.py
index 95265a958c35d..47ebfe311d9ea 100644
--- a/pandas/tests/arrays/test_timedeltas.py
+++ b/pandas/tests/arrays/test_timedeltas.py
@@ -290,8 +290,18 @@ def test_sum_2d_skipna_false(self):
         )._values
         tm.assert_timedelta_array_equal(result, expected)
 
-    def test_std(self):
-        tdi = pd.TimedeltaIndex(["0H", "4H", "NaT", "4H", "0H", "2H"])
+    # Adding a Timestamp makes this a test for DatetimeArray.std
+    @pytest.mark.parametrize(
+        "add",
+        [
+            pd.Timedelta(0),
+            pd.Timestamp.now(),
+            pd.Timestamp.now("UTC"),
+            pd.Timestamp.now("Asia/Tokyo"),
+        ],
+    )
+    def test_std(self, add):
+        tdi = pd.TimedeltaIndex(["0H", "4H", "NaT", "4H", "0H", "2H"]) + add
         arr = tdi.array
 
         result = arr.std(skipna=True)
@@ -303,9 +313,10 @@ def test_std(self):
         assert isinstance(result, pd.Timedelta)
         assert result == expected
 
-        result = nanops.nanstd(np.asarray(arr), skipna=True)
-        assert isinstance(result, pd.Timedelta)
-        assert result == expected
+        if getattr(arr, "tz", None) is None:
+            result = nanops.nanstd(np.asarray(arr), skipna=True)
+            assert isinstance(result, pd.Timedelta)
+            assert result == expected
 
         result = arr.std(skipna=False)
         assert result is pd.NaT
@@ -313,8 +324,9 @@ def test_std(self):
         result = tdi.std(skipna=False)
         assert result is pd.NaT
 
-        result = nanops.nanstd(np.asarray(arr), skipna=False)
-        assert result is pd.NaT
+        if getattr(arr, "tz", None) is None:
+            result = nanops.nanstd(np.asarray(arr), skipna=False)
+            assert result is pd.NaT
 
     def test_median(self):
         tdi = pd.TimedeltaIndex(["0H", "3H", "NaT", "5H06m", "0H", "2H"])
diff --git a/pandas/tests/reductions/test_stat_reductions.py b/pandas/tests/reductions/test_stat_reductions.py
index fd2746672a0eb..67e871f8b67c2 100644
--- a/pandas/tests/reductions/test_stat_reductions.py
+++ b/pandas/tests/reductions/test_stat_reductions.py
@@ -96,7 +96,7 @@ def _check_stat_op(
             string_series_[5:15] = np.NaN
 
             # mean, idxmax, idxmin, min, and max are valid for dates
-            if name not in ["max", "min", "mean", "median"]:
+            if name not in ["max", "min", "mean", "median", "std"]:
                 ds = Series(pd.date_range("1/1/2001", periods=10))
                 with pytest.raises(TypeError):
                     f(ds)