diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index 706dcaa34366a..a300736f948c1 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -100,6 +100,7 @@ Deprecations
 ~~~~~~~~~~~~
 - Deprecated silently dropping unrecognized timezones when parsing strings to datetimes (:issue:`18702`)
 - Deprecated :meth:`DataFrame._data` and :meth:`Series._data`, use public APIs instead (:issue:`33333`)
+- Deprecated :meth:`.Groupby.all` and :meth:`.GroupBy.any` with datetime64 or :class:`PeriodDtype` values, matching the :class:`Series` and :class:`DataFrame` deprecations (:issue:`34479`)
 - Deprecating pinning ``group.name`` to each group in :meth:`SeriesGroupBy.aggregate` aggregations; if your operation requires utilizing the groupby keys, iterate over the groupby object instead (:issue:`41090`)
 - Deprecated the default of ``observed=False`` in :meth:`DataFrame.groupby` and :meth:`Series.groupby`; this will default to ``True`` in a future version (:issue:`43999`)
 - Deprecated :meth:`DataFrameGroupBy.dtypes`, check ``dtypes`` on the underlying object instead (:issue:`51045`)
@@ -226,6 +227,7 @@ Groupby/resample/rolling
   grouped :class:`Series` or :class:`DataFrame` was a :class:`DatetimeIndex`, :class:`TimedeltaIndex`
   or :class:`PeriodIndex`, and the ``groupby`` method was given a function as its first argument,
   the function operated on the whole index rather than each element of the index. (:issue:`51979`)
+- Bug in :meth:`GroupBy.var` failing to raise ``TypeError`` when called with datetime64 or :class:`PeriodDtype` values (:issue:`52128`)
 -
 
 Reshaping
diff --git a/pandas/core/groupby/groupby.py b/pandas/core/groupby/groupby.py
index 12ecf5d54b356..e84a23be6c5bb 100644
--- a/pandas/core/groupby/groupby.py
+++ b/pandas/core/groupby/groupby.py
@@ -1500,13 +1500,7 @@ def _agg_py_fallback(
         # TODO: Is this exactly right; see WrappedCythonOp get_result_dtype?
         res_values = self.grouper.agg_series(ser, alt, preserve_dtype=True)
 
-        if isinstance(values, Categorical):
-            # Because we only get here with known dtype-preserving
-            #  reductions, we cast back to Categorical.
-            # TODO: if we ever get "rank" working, exclude it here.
-            res_values = type(values)._from_sequence(res_values, dtype=values.dtype)
-
-        elif ser.dtype == object:
+        if ser.dtype == object:
             res_values = res_values.astype(object, copy=False)
 
         # If we are DataFrameGroupBy and went through a SeriesGroupByPath
@@ -1544,8 +1538,8 @@ def array_func(values: ArrayLike) -> ArrayLike:
                 # and non-applicable functions
                 # try to python agg
                 # TODO: shouldn't min_count matter?
-                if how in ["any", "all"]:
-                    raise  # TODO: re-raise as TypeError?
+                if how in ["any", "all", "std", "sem"]:
+                    raise  # TODO: re-raise as TypeError?  should not be reached
                 result = self._agg_py_fallback(values, ndim=data.ndim, alt=alt)
 
             return result
diff --git a/pandas/core/groupby/ops.py b/pandas/core/groupby/ops.py
index 4f21d90ac5116..dad188e2d9304 100644
--- a/pandas/core/groupby/ops.py
+++ b/pandas/core/groupby/ops.py
@@ -18,6 +18,7 @@
     Sequence,
     final,
 )
+import warnings
 
 import numpy as np
 
@@ -37,6 +38,7 @@
 )
 from pandas.errors import AbstractMethodError
 from pandas.util._decorators import cache_readonly
+from pandas.util._exceptions import find_stack_level
 
 from pandas.core.dtypes.cast import (
     maybe_cast_pointwise_result,
@@ -258,19 +260,38 @@ def _disallow_invalid_ops(self, dtype: DtypeObj, is_numeric: bool = False):
                 #  case that would fail to raise
                 raise TypeError(f"Cannot perform {how} with non-ordered Categorical")
             if how not in ["rank", "any", "all", "first", "last", "min", "max"]:
-                # only "rank" is implemented in cython
-                raise NotImplementedError(f"{dtype} dtype not supported")
+                if self.kind == "transform":
+                    raise TypeError(f"{dtype} type does not support {how} operations")
+                raise TypeError(f"{dtype} dtype does not support aggregation '{how}'")
 
         elif is_sparse(dtype):
             raise NotImplementedError(f"{dtype} dtype not supported")
         elif is_datetime64_any_dtype(dtype):
             # Adding/multiplying datetimes is not valid
-            if how in ["sum", "prod", "cumsum", "cumprod"]:
+            if how in ["sum", "prod", "cumsum", "cumprod", "var"]:
                 raise TypeError(f"datetime64 type does not support {how} operations")
+            if how in ["any", "all"]:
+                # GH#34479
+                warnings.warn(
+                    f"'{how}' with datetime64 dtypes is deprecated and will raise in a "
+                    f"future version. Use (obj != pd.Timestamp(0)).{how}() instead.",
+                    FutureWarning,
+                    stacklevel=find_stack_level(),
+                )
+
         elif is_period_dtype(dtype):
             # Adding/multiplying Periods is not valid
-            if how in ["sum", "prod", "cumsum", "cumprod"]:
+            if how in ["sum", "prod", "cumsum", "cumprod", "var"]:
                 raise TypeError(f"Period type does not support {how} operations")
+            if how in ["any", "all"]:
+                # GH#34479
+                warnings.warn(
+                    f"'{how}' with PeriodDtype is deprecated and will raise in a "
+                    f"future version. Use (obj != pd.Period(0, freq)).{how}() instead.",
+                    FutureWarning,
+                    stacklevel=find_stack_level(),
+                )
+
         elif is_timedelta64_dtype(dtype):
             # timedeltas we can add but not multiply
             if how in ["prod", "cumprod"]:
diff --git a/pandas/tests/groupby/test_raises.py b/pandas/tests/groupby/test_raises.py
index 60d4f98aa33f7..bd3686354e432 100644
--- a/pandas/tests/groupby/test_raises.py
+++ b/pandas/tests/groupby/test_raises.py
@@ -231,16 +231,23 @@ def test_groupby_raises_datetime(how, by, groupby_series, groupby_func):
         "skew": (TypeError, r"dtype datetime64\[ns\] does not support reduction"),
         "std": (None, ""),
         "sum": (TypeError, "datetime64 type does not support sum operations"),
-        "var": (None, ""),
+        "var": (TypeError, "datetime64 type does not support var operations"),
     }[groupby_func]
 
     if klass is None:
-        if how == "method":
-            getattr(gb, groupby_func)(*args)
-        elif how == "agg":
-            gb.agg(groupby_func, *args)
-        else:
-            gb.transform(groupby_func, *args)
+        warn = None
+        warn_msg = f"'{groupby_func}' with datetime64 dtypes is deprecated"
+        if groupby_func in ["any", "all"]:
+            warn = FutureWarning
+
+        with tm.assert_produces_warning(warn, match=warn_msg):
+            if how == "method":
+                getattr(gb, groupby_func)(*args)
+            elif how == "agg":
+                gb.agg(groupby_func, *args)
+            else:
+                gb.transform(groupby_func, *args)
+
     else:
         with pytest.raises(klass, match=msg):
             if how == "method":
@@ -383,11 +390,21 @@ def test_groupby_raises_category(
         "max": (None, ""),
         "mean": (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'mean'",
+            "|".join(
+                [
+                    "'Categorical' .* does not support reduction 'mean'",
+                    "category dtype does not support aggregation 'mean'",
+                ]
+            ),
         ),
         "median": (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'median'",
+            "|".join(
+                [
+                    "'Categorical' .* does not support reduction 'median'",
+                    "category dtype does not support aggregation 'median'",
+                ]
+            ),
         ),
         "min": (None, ""),
         "ngroup": (None, ""),
@@ -401,7 +418,12 @@ def test_groupby_raises_category(
         "rank": (None, ""),
         "sem": (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'sem'",
+            "|".join(
+                [
+                    "'Categorical' .* does not support reduction 'sem'",
+                    "category dtype does not support aggregation 'sem'",
+                ]
+            ),
         ),
         "shift": (None, ""),
         "size": (None, ""),
@@ -411,12 +433,22 @@ def test_groupby_raises_category(
         ),
         "std": (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'std'",
+            "|".join(
+                [
+                    "'Categorical' .* does not support reduction 'std'",
+                    "category dtype does not support aggregation 'std'",
+                ]
+            ),
         ),
         "sum": (TypeError, "category type does not support sum operations"),
         "var": (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'var'",
+            "|".join(
+                [
+                    "'Categorical' .* does not support reduction 'var'",
+                    "category dtype does not support aggregation 'var'",
+                ]
+            ),
         ),
     }[groupby_func]
 
@@ -489,7 +521,7 @@ def test_groupby_raises_category_np(how, by, groupby_series, groupby_func_np):
         np.sum: (TypeError, "category type does not support sum operations"),
         np.mean: (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'mean'",
+            "category dtype does not support aggregation 'mean'",
         ),
     }[groupby_func_np]
 
@@ -585,14 +617,8 @@ def test_groupby_raises_category_on_category(
         else (None, ""),
         "last": (None, ""),
         "max": (None, ""),
-        "mean": (
-            TypeError,
-            "'Categorical' with dtype category does not support reduction 'mean'",
-        ),
-        "median": (
-            TypeError,
-            "'Categorical' with dtype category does not support reduction 'median'",
-        ),
+        "mean": (TypeError, "category dtype does not support aggregation 'mean'"),
+        "median": (TypeError, "category dtype does not support aggregation 'median'"),
         "min": (None, ""),
         "ngroup": (None, ""),
         "nunique": (None, ""),
@@ -602,7 +628,12 @@ def test_groupby_raises_category_on_category(
         "rank": (None, ""),
         "sem": (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'sem'",
+            "|".join(
+                [
+                    "'Categorical' .* does not support reduction 'sem'",
+                    "category dtype does not support aggregation 'sem'",
+                ]
+            ),
         ),
         "shift": (None, ""),
         "size": (None, ""),
@@ -612,12 +643,22 @@ def test_groupby_raises_category_on_category(
         ),
         "std": (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'std'",
+            "|".join(
+                [
+                    "'Categorical' .* does not support reduction 'std'",
+                    "category dtype does not support aggregation 'std'",
+                ]
+            ),
         ),
         "sum": (TypeError, "category type does not support sum operations"),
         "var": (
             TypeError,
-            "'Categorical' with dtype category does not support reduction 'var'",
+            "|".join(
+                [
+                    "'Categorical' .* does not support reduction 'var'",
+                    "category dtype does not support aggregation 'var'",
+                ]
+            ),
         ),
     }[groupby_func]