diff --git a/doc/source/whatsnew/v1.0.0.rst b/doc/source/whatsnew/v1.0.0.rst
index ec6ad38bbc7cf..2f6d502901af1 100755
--- a/doc/source/whatsnew/v1.0.0.rst
+++ b/doc/source/whatsnew/v1.0.0.rst
@@ -246,6 +246,44 @@ source, you should no longer need to install Cython into your build environment
 Backwards incompatible API changes
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
+.. _whatsnew_100.api_breaking.nat_sort:
+
+Changed sort position for ``NaT``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+:attr:`NaT` will now sort at the *end* rather than the beginning in sorting functions (:issue:`29884`).
+This matches the behavior in NumPy 1.18 and newer, which makes the ``NaT`` behavior consistent
+with other missing values like :attr:`numpy.nan`.
+
+.. ipython:: python
+
+   values = pd.Index(['2001', 'NaT', '2000'], dtype='datetime64[ns]')
+
+*pandas 0.25.x*
+
+.. code-block:: python
+
+   >>> values.sort_values()
+   DatetimeIndex(['NaT', '2000-01-01', '2001-01-01'], dtype='datetime64[ns]', freq=None)
+
+   >>> values.argsort()
+   array([1, 2, 0])
+
+
+*pandas 1.0.0*
+
+.. ipython:: python
+
+   values.sort_values()
+   values.argsort()
+
+This affects all sorting functions on indexes, Series, DataFrames, and arrays.
+
+
+.. note::
+
+   This change was made between pandas 1.0.0rc0 and pandas 1.0.0.
+
 .. _whatsnew_100.api_breaking.MultiIndex._names:
 
 Avoid using names from ``MultiIndex.levels``
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index 70637026c278d..d9b0ee0fc0ba0 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -710,7 +710,7 @@ def _from_factorized(cls, values, original):
         return cls(values, dtype=original.dtype)
 
     def _values_for_argsort(self):
-        return self._data
+        return self._data.view("M8[ns]")
 
     # ------------------------------------------------------------------
     # Additional array methods
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 1bfec9fbad0ed..ea89fb38e8977 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -191,10 +191,8 @@ def sort_values(self, return_indexer=False, ascending=True):
             sorted_index = self.take(_as)
             return sorted_index, _as
         else:
-            # NB: using asi8 instead of _ndarray_values matters in numpy 1.18
-            #  because the treatment of NaT has been changed to put NaT last
-            #  instead of first.
-            sorted_values = np.sort(self.asi8)
+            values = self._data
+            sorted_values = np.sort(values.view("M8[ns]")).view("i8")
 
             freq = self.freq
             if freq is not None and not is_period_dtype(self):
@@ -224,6 +222,9 @@ def take(self, indices, axis=0, allow_fill=True, fill_value=None, **kwargs):
             self, indices, axis, allow_fill, fill_value, **kwargs
         )
 
+    def argsort(self, *args, **kwargs) -> np.ndarray:
+        return np.argsort(self._data, *args, **kwargs)
+
     @Appender(_shared_docs["searchsorted"])
     def searchsorted(self, value, side="left", sorter=None):
         if isinstance(value, str):
diff --git a/pandas/tests/arrays/test_datetimelike.py b/pandas/tests/arrays/test_datetimelike.py
index 87b825c8c27bd..329311643c7f8 100644
--- a/pandas/tests/arrays/test_datetimelike.py
+++ b/pandas/tests/arrays/test_datetimelike.py
@@ -795,3 +795,11 @@ def test_to_numpy_extra(array):
     assert result[0] == result[1]
 
     tm.assert_equal(array, original)
+
+
+@pytest.mark.parametrize("dtype", ["datetime64[ns]", "timedelta64[ns]", "Period[D]"])
+def test_argsort(dtype):
+    a = pd.array([2001, pd.NaT, 2000], dtype=dtype)
+    result = a.argsort()
+    expected = np.array([2, 0, 1])
+    tm.assert_numpy_array_equal(result, expected)
diff --git a/pandas/tests/indexes/datetimes/test_ops.py b/pandas/tests/indexes/datetimes/test_ops.py
index ecd4ace705e9e..1f59cf428748d 100644
--- a/pandas/tests/indexes/datetimes/test_ops.py
+++ b/pandas/tests/indexes/datetimes/test_ops.py
@@ -247,7 +247,7 @@ def test_order_with_freq(self, idx):
             ),
             (
                 [pd.NaT, "2011-01-03", "2011-01-05", "2011-01-02", pd.NaT],
-                [pd.NaT, pd.NaT, "2011-01-02", "2011-01-03", "2011-01-05"],
+                ["2011-01-02", "2011-01-03", "2011-01-05", pd.NaT, pd.NaT],
             ),
         ],
     )
@@ -269,14 +269,17 @@ def test_order_without_freq(self, index_dates, expected_dates, tz_naive_fixture)
         ordered, indexer = index.sort_values(return_indexer=True)
         tm.assert_index_equal(ordered, expected)
 
-        exp = np.array([0, 4, 3, 1, 2])
+        if index.isna().any():
+            exp = np.array([3, 1, 2, 0, 4])
+        else:
+            exp = np.array([0, 4, 3, 1, 2])
         tm.assert_numpy_array_equal(indexer, exp, check_dtype=False)
         assert ordered.freq is None
 
         ordered, indexer = index.sort_values(return_indexer=True, ascending=False)
         tm.assert_index_equal(ordered, expected[::-1])
 
-        exp = np.array([2, 1, 3, 4, 0])
+        exp = exp[::-1]
         tm.assert_numpy_array_equal(indexer, exp, check_dtype=False)
         assert ordered.freq is None