diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index f751a91cecf19..49f617501980b 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -322,6 +322,10 @@ Other API changes
 ^^^^^^^^^^^^^^^^^
 
 - Sorting in descending order is now stable for :meth:`Series.sort_values` and :meth:`Index.sort_values` for DateTime-like :class:`Index` subclasses. This will affect sort order when sorting :class:`DataFrame` on multiple columns, sorting with a key function that produces duplicates, or requesting the sorting index when using :meth:`Index.sort_values`. When using :meth:`Series.value_counts`, count of missing values is no longer the last in the list of duplicate counts, and its position corresponds to the position in the original :class:`Series`. When using :meth:`Index.sort_values` for DateTime-like :class:`Index` subclasses, NaTs ignored the ``na_position`` argument and were sorted to the beggining. Now they respect ``na_position``, the default being ``last``, same as other :class:`Index` subclasses. (:issue:`35992`)
+- Passing an invalid ``fill_value`` to :meth:`Categorical.take`, :meth:`DatetimeArray.take`, :meth:`TimedeltaArray.take`, :meth:`PeriodArray.take` now raises ``TypeError`` instead of ``ValueError`` (:issue:`37733`)
+- Passing an invalid ``fill_value`` to :meth:`Series.shift` with a ``CategoricalDtype`` now raises ``TypeError`` instead of ``ValueError`` (:issue:`37733`)
+- Passing an invalid value to :meth:`IntervalIndex.insert` or :meth:`CategoricalIndex.insert` now raises a ``TypeError`` instead of a ``ValueError`` (:issue:`37733`)
+- Attempting to reindex a :class:`Series` with a :class:`CategoricalIndex` with an invalid ``fill_value`` now raises ``TypeError`` instead of ``ValueError`` (:issue:`37733`)
 
 .. ---------------------------------------------------------------------------
 
diff --git a/pandas/core/arrays/_mixins.py b/pandas/core/arrays/_mixins.py
index d84e2e2ad295b..ddcf225d3585f 100644
--- a/pandas/core/arrays/_mixins.py
+++ b/pandas/core/arrays/_mixins.py
@@ -74,7 +74,7 @@ def take(
     def _validate_fill_value(self, fill_value):
         """
         If a fill_value is passed to `take` convert it to a representation
-        suitable for self._ndarray, raising ValueError if this is not possible.
+        suitable for self._ndarray, raising TypeError if this is not possible.
 
         Parameters
         ----------
@@ -86,7 +86,7 @@ def _validate_fill_value(self, fill_value):
 
         Raises
         ------
-        ValueError
+        TypeError
         """
         raise AbstractMethodError(self)
 
diff --git a/pandas/core/arrays/categorical.py b/pandas/core/arrays/categorical.py
index 9f011bc9d2651..970163df908ec 100644
--- a/pandas/core/arrays/categorical.py
+++ b/pandas/core/arrays/categorical.py
@@ -1190,7 +1190,7 @@ def _validate_searchsorted_value(self, value):
     def _validate_fill_value(self, fill_value):
         """
         Convert a user-facing fill_value to a representation to use with our
-        underlying ndarray, raising ValueError if this is not possible.
+        underlying ndarray, raising TypeError if this is not possible.
 
         Parameters
         ----------
@@ -1202,7 +1202,7 @@ def _validate_fill_value(self, fill_value):
 
         Raises
         ------
-        ValueError
+        TypeError
         """
 
         if is_valid_nat_for_dtype(fill_value, self.categories.dtype):
@@ -1210,7 +1210,7 @@ def _validate_fill_value(self, fill_value):
         elif fill_value in self.categories:
             fill_value = self._unbox_scalar(fill_value)
         else:
-            raise ValueError(
+            raise TypeError(
                 f"'fill_value={fill_value}' is not present "
                 "in this Categorical's categories"
             )
@@ -1659,7 +1659,6 @@ def fillna(self, value=None, method=None, limit=None):
                 # We get ndarray or Categorical if called via Series.fillna,
                 #  where it will unwrap another aligned Series before getting here
                 codes[mask] = new_codes[mask]
-
             else:
                 codes[mask] = new_codes
 
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index f2f843886e802..d81a4567732bc 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -462,7 +462,7 @@ def _validate_comparison_value(self, other):
     def _validate_fill_value(self, fill_value):
         """
         If a fill_value is passed to `take` convert it to an i8 representation,
-        raising ValueError if this is not possible.
+        raising TypeError if this is not possible.
 
         Parameters
         ----------
@@ -474,19 +474,9 @@ def _validate_fill_value(self, fill_value):
 
         Raises
         ------
-        ValueError
+        TypeError
         """
-        msg = (
-            f"'fill_value' should be a {self._scalar_type}. "
-            f"Got '{str(fill_value)}'."
-        )
-        try:
-            return self._validate_scalar(fill_value)
-        except TypeError as err:
-            if "Cannot compare tz-naive and tz-aware" in str(err):
-                # tzawareness-compat
-                raise
-            raise ValueError(msg) from err
+        return self._validate_scalar(fill_value)
 
     def _validate_shift_value(self, fill_value):
         # TODO(2.0): once this deprecation is enforced, use _validate_fill_value
diff --git a/pandas/core/arrays/interval.py b/pandas/core/arrays/interval.py
index a2eb506c6747a..977e4abff4287 100644
--- a/pandas/core/arrays/interval.py
+++ b/pandas/core/arrays/interval.py
@@ -649,7 +649,7 @@ def fillna(self, value=None, method=None, limit=None):
         if limit is not None:
             raise TypeError("limit is not supported for IntervalArray.")
 
-        value_left, value_right = self._validate_fillna_value(value)
+        value_left, value_right = self._validate_fill_value(value)
 
         left = self.left.fillna(value=value_left)
         right = self.right.fillna(value=value_right)
@@ -870,7 +870,7 @@ def _validate_scalar(self, value):
             # GH#18295
             left = right = value
         else:
-            raise ValueError(
+            raise TypeError(
                 "can only insert Interval objects and NA into an IntervalArray"
             )
         return left, right
@@ -878,17 +878,6 @@ def _validate_scalar(self, value):
     def _validate_fill_value(self, value):
         return self._validate_scalar(value)
 
-    def _validate_fillna_value(self, value):
-        # This mirrors Datetimelike._validate_fill_value
-        try:
-            return self._validate_scalar(value)
-        except ValueError as err:
-            msg = (
-                "'IntervalArray.fillna' only supports filling with a "
-                f"scalar 'pandas.Interval'. Got a '{type(value).__name__}' instead."
-            )
-            raise TypeError(msg) from err
-
     def _validate_setitem_value(self, value):
         needs_float_conversion = False
 
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 5a3f2b0853c4f..5790c6db6405f 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -3660,7 +3660,12 @@ def insert(self, loc: int, item):
                 # must insert at end otherwise you have to recompute all the
                 # other codes
                 lev_loc = len(level)
-                level = level.insert(lev_loc, k)
+                try:
+                    level = level.insert(lev_loc, k)
+                except TypeError:
+                    # TODO: Should this be done inside insert?
+                    # TODO: smarter casting rules?
+                    level = level.astype(object).insert(lev_loc, k)
             else:
                 lev_loc = level.get_loc(k)
 
diff --git a/pandas/tests/arrays/categorical/test_take.py b/pandas/tests/arrays/categorical/test_take.py
index 7a27f5c3e73ad..97d9db483c401 100644
--- a/pandas/tests/arrays/categorical/test_take.py
+++ b/pandas/tests/arrays/categorical/test_take.py
@@ -79,7 +79,7 @@ def test_take_fill_value_new_raises(self):
         # https://github.com/pandas-dev/pandas/issues/23296
         cat = Categorical(["a", "b", "c"])
         xpr = r"'fill_value=d' is not present in this Categorical's categories"
-        with pytest.raises(ValueError, match=xpr):
+        with pytest.raises(TypeError, match=xpr):
             cat.take([0, 1, -1], fill_value="d", allow_fill=True)
 
     def test_take_nd_deprecated(self):
diff --git a/pandas/tests/arrays/test_datetimelike.py b/pandas/tests/arrays/test_datetimelike.py
index ec20c829f1544..c24f789b30313 100644
--- a/pandas/tests/arrays/test_datetimelike.py
+++ b/pandas/tests/arrays/test_datetimelike.py
@@ -145,8 +145,8 @@ def test_take_fill_raises(self, fill_value):
 
         arr = self.array_cls._simple_new(data, freq="D")
 
-        msg = f"'fill_value' should be a {self.dtype}. Got '{fill_value}'"
-        with pytest.raises(ValueError, match=msg):
+        msg = f"value should be a '{arr._scalar_type.__name__}' or 'NaT'. Got"
+        with pytest.raises(TypeError, match=msg):
             arr.take([0, 1], allow_fill=True, fill_value=fill_value)
 
     def test_take_fill(self):
@@ -169,8 +169,8 @@ def test_take_fill_str(self, arr1d):
         expected = arr1d[[-1, 1]]
         tm.assert_equal(result, expected)
 
-        msg = r"'fill_value' should be a <.*>\. Got 'foo'"
-        with pytest.raises(ValueError, match=msg):
+        msg = f"value should be a '{arr1d._scalar_type.__name__}' or 'NaT'. Got"
+        with pytest.raises(TypeError, match=msg):
             arr1d.take([-1, 1], allow_fill=True, fill_value="foo")
 
     def test_concat_same_type(self):
@@ -745,13 +745,12 @@ def test_take_fill_valid(self, arr1d):
         result = arr.take([-1, 1], allow_fill=True, fill_value=now)
         assert result[0] == now
 
-        msg = f"'fill_value' should be a {self.dtype}. Got '0 days 00:00:00'."
-        with pytest.raises(ValueError, match=msg):
+        msg = f"value should be a '{arr1d._scalar_type.__name__}' or 'NaT'. Got"
+        with pytest.raises(TypeError, match=msg):
             # fill_value Timedelta invalid
             arr.take([-1, 1], allow_fill=True, fill_value=now - now)
 
-        msg = f"'fill_value' should be a {self.dtype}. Got '2014Q1'."
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             # fill_value Period invalid
             arr.take([-1, 1], allow_fill=True, fill_value=pd.Period("2014Q1"))
 
@@ -763,14 +762,13 @@ def test_take_fill_valid(self, arr1d):
             arr.take([-1, 1], allow_fill=True, fill_value=now)
 
         value = pd.NaT.value
-        msg = f"'fill_value' should be a {self.dtype}. Got '{value}'."
-        with pytest.raises(ValueError, match=msg):
+        msg = f"value should be a '{arr1d._scalar_type.__name__}' or 'NaT'. Got"
+        with pytest.raises(TypeError, match=msg):
             # require NaT, not iNaT, as it could be confused with an integer
             arr.take([-1, 1], allow_fill=True, fill_value=value)
 
         value = np.timedelta64("NaT", "ns")
-        msg = f"'fill_value' should be a {self.dtype}. Got '{str(value)}'."
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             # require appropriate-dtype if we have a NA value
             arr.take([-1, 1], allow_fill=True, fill_value=value)
 
@@ -932,20 +930,18 @@ def test_take_fill_valid(self, timedelta_index):
 
         now = pd.Timestamp.now()
         value = now
-        msg = f"'fill_value' should be a {self.dtype}. Got '{value}'."
-        with pytest.raises(ValueError, match=msg):
+        msg = f"value should be a '{arr._scalar_type.__name__}' or 'NaT'. Got"
+        with pytest.raises(TypeError, match=msg):
             # fill_value Timestamp invalid
             arr.take([0, 1], allow_fill=True, fill_value=value)
 
         value = now.to_period("D")
-        msg = f"'fill_value' should be a {self.dtype}. Got '{value}'."
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             # fill_value Period invalid
             arr.take([0, 1], allow_fill=True, fill_value=value)
 
         value = np.datetime64("NaT", "ns")
-        msg = f"'fill_value' should be a {self.dtype}. Got '{str(value)}'."
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             # require appropriate-dtype if we have a NA value
             arr.take([-1, 1], allow_fill=True, fill_value=value)
 
@@ -981,14 +977,13 @@ def test_take_fill_valid(self, arr1d):
         arr = arr1d
 
         value = pd.NaT.value
-        msg = f"'fill_value' should be a {self.dtype}. Got '{value}'."
-        with pytest.raises(ValueError, match=msg):
+        msg = f"value should be a '{arr1d._scalar_type.__name__}' or 'NaT'. Got"
+        with pytest.raises(TypeError, match=msg):
             # require NaT, not iNaT, as it could be confused with an integer
             arr.take([-1, 1], allow_fill=True, fill_value=value)
 
         value = np.timedelta64("NaT", "ns")
-        msg = f"'fill_value' should be a {self.dtype}. Got '{str(value)}'."
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             # require appropriate-dtype if we have a NA value
             arr.take([-1, 1], allow_fill=True, fill_value=value)
 
diff --git a/pandas/tests/arrays/test_period.py b/pandas/tests/arrays/test_period.py
index 0d81e8e733842..f96a15d5b2e7c 100644
--- a/pandas/tests/arrays/test_period.py
+++ b/pandas/tests/arrays/test_period.py
@@ -113,7 +113,8 @@ def test_take_raises():
     with pytest.raises(IncompatibleFrequency, match="freq"):
         arr.take([0, -1], allow_fill=True, fill_value=pd.Period("2000", freq="W"))
 
-    with pytest.raises(ValueError, match="foo"):
+    msg = "value should be a 'Period' or 'NaT'. Got 'str' instead"
+    with pytest.raises(TypeError, match=msg):
         arr.take([0, -1], allow_fill=True, fill_value="foo")
 
 
diff --git a/pandas/tests/extension/test_interval.py b/pandas/tests/extension/test_interval.py
index 4fdcf930d224f..ec834118ea7a1 100644
--- a/pandas/tests/extension/test_interval.py
+++ b/pandas/tests/extension/test_interval.py
@@ -136,8 +136,8 @@ def test_fillna_limit_backfill(self):
     def test_fillna_series(self):
         pass
 
-    def test_non_scalar_raises(self, data_missing):
-        msg = "Got a 'list' instead."
+    def test_fillna_non_scalar_raises(self, data_missing):
+        msg = "can only insert Interval objects and NA into an IntervalArray"
         with pytest.raises(TypeError, match=msg):
             data_missing.fillna([1, 1])
 
diff --git a/pandas/tests/frame/test_stack_unstack.py b/pandas/tests/frame/test_stack_unstack.py
index d1425c85caaee..3fa17c1764de3 100644
--- a/pandas/tests/frame/test_stack_unstack.py
+++ b/pandas/tests/frame/test_stack_unstack.py
@@ -245,7 +245,7 @@ def test_unstack_fill_frame_categorical(self):
 
         # Fill with non-category results in a ValueError
         msg = r"'fill_value=d' is not present in"
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             data.unstack(fill_value="d")
 
         # Fill with category value replaces missing values as expected
diff --git a/pandas/tests/indexes/categorical/test_category.py b/pandas/tests/indexes/categorical/test_category.py
index 324a2535bc465..1a05dbe2bb230 100644
--- a/pandas/tests/indexes/categorical/test_category.py
+++ b/pandas/tests/indexes/categorical/test_category.py
@@ -172,7 +172,7 @@ def test_insert(self):
 
         # invalid
         msg = "'fill_value=d' is not present in this Categorical's categories"
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             ci.insert(0, "d")
 
         # GH 18295 (test missing)
@@ -184,7 +184,7 @@ def test_insert(self):
     def test_insert_na_mismatched_dtype(self):
         ci = CategoricalIndex([0, 1, 1])
         msg = "'fill_value=NaT' is not present in this Categorical's categories"
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             ci.insert(0, pd.NaT)
 
     def test_delete(self):
diff --git a/pandas/tests/indexes/categorical/test_reindex.py b/pandas/tests/indexes/categorical/test_reindex.py
index 189582ea635d2..668c559abd08e 100644
--- a/pandas/tests/indexes/categorical/test_reindex.py
+++ b/pandas/tests/indexes/categorical/test_reindex.py
@@ -57,5 +57,5 @@ def test_reindex_missing_category(self):
         # GH: 18185
         ser = Series([1, 2, 3, 1], dtype="category")
         msg = "'fill_value=-1' is not present in this Categorical's categories"
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             ser.reindex([1, 2, 3, 4, 5], fill_value=-1)
diff --git a/pandas/tests/indexes/interval/test_interval.py b/pandas/tests/indexes/interval/test_interval.py
index 157446b1fff5d..45683ba48b4c4 100644
--- a/pandas/tests/indexes/interval/test_interval.py
+++ b/pandas/tests/indexes/interval/test_interval.py
@@ -192,7 +192,7 @@ def test_insert(self, data):
 
         # invalid type
         msg = "can only insert Interval objects and NA into an IntervalArray"
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             data.insert(1, "foo")
 
         # invalid closed
@@ -213,7 +213,7 @@ def test_insert(self, data):
         if data.left.dtype.kind not in ["m", "M"]:
             # trying to insert pd.NaT into a numeric-dtyped Index should cast/raise
             msg = "can only insert Interval objects and NA into an IntervalArray"
-            with pytest.raises(ValueError, match=msg):
+            with pytest.raises(TypeError, match=msg):
                 result = data.insert(1, pd.NaT)
         else:
             result = data.insert(1, pd.NaT)
diff --git a/pandas/tests/indexing/test_categorical.py b/pandas/tests/indexing/test_categorical.py
index 20d7662855ab3..79e9df8ce30cb 100644
--- a/pandas/tests/indexing/test_categorical.py
+++ b/pandas/tests/indexing/test_categorical.py
@@ -60,9 +60,9 @@ def test_loc_scalar(self):
             df.loc["d"] = 10
 
         msg = "'fill_value=d' is not present in this Categorical's categories"
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             df.loc["d", "A"] = 10
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             df.loc["d", "C"] = 10
 
         with pytest.raises(KeyError, match="^1$"):
diff --git a/pandas/tests/series/methods/test_shift.py b/pandas/tests/series/methods/test_shift.py
index d38d70abba923..60ec0a90e906f 100644
--- a/pandas/tests/series/methods/test_shift.py
+++ b/pandas/tests/series/methods/test_shift.py
@@ -170,7 +170,7 @@ def test_shift_categorical_fill_value(self):
 
         # check for incorrect fill_value
         msg = "'fill_value=f' is not present in this Categorical's categories"
-        with pytest.raises(ValueError, match=msg):
+        with pytest.raises(TypeError, match=msg):
             ts.shift(1, fill_value="f")
 
     def test_shift_dst(self):