diff --git a/doc/source/whatsnew/v1.4.0.rst b/doc/source/whatsnew/v1.4.0.rst
index ab3987c8c74a4..916bcf3db9a4a 100644
--- a/doc/source/whatsnew/v1.4.0.rst
+++ b/doc/source/whatsnew/v1.4.0.rst
@@ -219,7 +219,7 @@ Other enhancements
 - :meth:`DataFrame.dropna` now accepts a single label as ``subset`` along with array-like (:issue:`41021`)
 - :class:`ExcelWriter` argument ``if_sheet_exists="overlay"`` option added (:issue:`40231`)
 - :meth:`read_excel` now accepts a ``decimal`` argument that allow the user to specify the decimal point when parsing string columns to numeric (:issue:`14403`)
-- :meth:`.GroupBy.mean` now supports `Numba <http://numba.pydata.org/>`_ execution with the ``engine`` keyword (:issue:`43731`)
+- :meth:`.GroupBy.mean`, :meth:`.GroupBy.std`, and :meth:`.GroupBy.var` now supports `Numba <http://numba.pydata.org/>`_ execution with the ``engine`` keyword (:issue:`43731`, :issue:`44862`)
 - :meth:`Timestamp.isoformat`, now handles the ``timespec`` argument from the base :class:``datetime`` class (:issue:`26131`)
 - :meth:`NaT.to_numpy` ``dtype`` argument is now respected, so ``np.timedelta64`` can be returned (:issue:`44460`)
 - New option ``display.max_dir_items`` customizes the number of columns added to :meth:`Dataframe.__dir__` and suggested for tab completion (:issue:`37996`)
diff --git a/pandas/core/groupby/groupby.py b/pandas/core/groupby/groupby.py
index 2876ec1cb5a0d..e11d420ada29f 100644
--- a/pandas/core/groupby/groupby.py
+++ b/pandas/core/groupby/groupby.py
@@ -1272,6 +1272,7 @@ def _numba_agg_general(
         func: Callable,
         engine_kwargs: dict[str, bool] | None,
         numba_cache_key_str: str,
+        *aggregator_args,
     ):
         """
         Perform groupby with a standard numerical aggregation function (e.g. mean)
@@ -1291,7 +1292,7 @@ def _numba_agg_general(
         aggregator = executor.generate_shared_aggregator(
             func, engine_kwargs, numba_cache_key_str
         )
-        result = aggregator(sorted_data, starts, ends, 0)
+        result = aggregator(sorted_data, starts, ends, 0, *aggregator_args)
 
         cache_key = (func, numba_cache_key_str)
         if cache_key not in NUMBA_FUNC_CACHE:
@@ -1989,7 +1990,12 @@ def median(self, numeric_only: bool | lib.NoDefault = lib.no_default):
     @final
     @Substitution(name="groupby")
     @Appender(_common_see_also)
-    def std(self, ddof: int = 1):
+    def std(
+        self,
+        ddof: int = 1,
+        engine: str | None = None,
+        engine_kwargs: dict[str, bool] | None = None,
+    ):
         """
         Compute standard deviation of groups, excluding missing values.
 
@@ -2000,23 +2006,52 @@ def std(self, ddof: int = 1):
         ddof : int, default 1
             Degrees of freedom.
 
+        engine : str, default None
+            * ``'cython'`` : Runs the operation through C-extensions from cython.
+            * ``'numba'`` : Runs the operation through JIT compiled code from numba.
+            * ``None`` : Defaults to ``'cython'`` or globally setting
+              ``compute.use_numba``
+
+            .. versionadded:: 1.4.0
+
+        engine_kwargs : dict, default None
+            * For ``'cython'`` engine, there are no accepted ``engine_kwargs``
+            * For ``'numba'`` engine, the engine can accept ``nopython``, ``nogil``
+              and ``parallel`` dictionary keys. The values must either be ``True`` or
+              ``False``. The default ``engine_kwargs`` for the ``'numba'`` engine is
+              ``{{'nopython': True, 'nogil': False, 'parallel': False}}``
+
+            .. versionadded:: 1.4.0
+
         Returns
         -------
         Series or DataFrame
             Standard deviation of values within each group.
         """
-        return self._get_cythonized_result(
-            libgroupby.group_var,
-            needs_counts=True,
-            cython_dtype=np.dtype(np.float64),
-            post_processing=lambda vals, inference: np.sqrt(vals),
-            ddof=ddof,
-        )
+        if maybe_use_numba(engine):
+            from pandas.core._numba.kernels import sliding_var
+
+            return np.sqrt(
+                self._numba_agg_general(sliding_var, engine_kwargs, "groupby_std", ddof)
+            )
+        else:
+            return self._get_cythonized_result(
+                libgroupby.group_var,
+                needs_counts=True,
+                cython_dtype=np.dtype(np.float64),
+                post_processing=lambda vals, inference: np.sqrt(vals),
+                ddof=ddof,
+            )
 
     @final
     @Substitution(name="groupby")
     @Appender(_common_see_also)
-    def var(self, ddof: int = 1):
+    def var(
+        self,
+        ddof: int = 1,
+        engine: str | None = None,
+        engine_kwargs: dict[str, bool] | None = None,
+    ):
         """
         Compute variance of groups, excluding missing values.
 
@@ -2027,20 +2062,46 @@ def var(self, ddof: int = 1):
         ddof : int, default 1
             Degrees of freedom.
 
+        engine : str, default None
+            * ``'cython'`` : Runs the operation through C-extensions from cython.
+            * ``'numba'`` : Runs the operation through JIT compiled code from numba.
+            * ``None`` : Defaults to ``'cython'`` or globally setting
+              ``compute.use_numba``
+
+            .. versionadded:: 1.4.0
+
+        engine_kwargs : dict, default None
+            * For ``'cython'`` engine, there are no accepted ``engine_kwargs``
+            * For ``'numba'`` engine, the engine can accept ``nopython``, ``nogil``
+              and ``parallel`` dictionary keys. The values must either be ``True`` or
+              ``False``. The default ``engine_kwargs`` for the ``'numba'`` engine is
+              ``{{'nopython': True, 'nogil': False, 'parallel': False}}``
+
+            .. versionadded:: 1.4.0
+
         Returns
         -------
         Series or DataFrame
             Variance of values within each group.
         """
-        if ddof == 1:
-            numeric_only = self._resolve_numeric_only(lib.no_default)
-            return self._cython_agg_general(
-                "var", alt=lambda x: Series(x).var(ddof=ddof), numeric_only=numeric_only
+        if maybe_use_numba(engine):
+            from pandas.core._numba.kernels import sliding_var
+
+            return self._numba_agg_general(
+                sliding_var, engine_kwargs, "groupby_var", ddof
             )
         else:
-            func = lambda x: x.var(ddof=ddof)
-            with self._group_selection_context():
-                return self._python_agg_general(func)
+            if ddof == 1:
+                numeric_only = self._resolve_numeric_only(lib.no_default)
+                return self._cython_agg_general(
+                    "var",
+                    alt=lambda x: Series(x).var(ddof=ddof),
+                    numeric_only=numeric_only,
+                )
+            else:
+                func = lambda x: x.var(ddof=ddof)
+                with self._group_selection_context():
+                    return self._python_agg_general(func)
 
     @final
     @Substitution(name="groupby")
diff --git a/pandas/tests/groupby/conftest.py b/pandas/tests/groupby/conftest.py
index b61f6700f8d1d..596e9e0a4de77 100644
--- a/pandas/tests/groupby/conftest.py
+++ b/pandas/tests/groupby/conftest.py
@@ -174,3 +174,17 @@ def nogil(request):
 def nopython(request):
     """nopython keyword argument for numba.jit"""
     return request.param
+
+
+@pytest.fixture(
+    params=[
+        ("mean", {}),
+        ("var", {"ddof": 1}),
+        ("var", {"ddof": 0}),
+        ("std", {"ddof": 1}),
+        ("std", {"ddof": 0}),
+    ]
+)
+def numba_supported_reductions(request):
+    """reductions supported with engine='numba'"""
+    return request.param
diff --git a/pandas/tests/groupby/test_numba.py b/pandas/tests/groupby/test_numba.py
index 20fd02b21a744..6fc1f0d808bb2 100644
--- a/pandas/tests/groupby/test_numba.py
+++ b/pandas/tests/groupby/test_numba.py
@@ -13,39 +13,55 @@
 @pytest.mark.filterwarnings("ignore:\n")
 # Filter warnings when parallel=True and the function can't be parallelized by Numba
 class TestEngine:
-    def test_cython_vs_numba_frame(self, sort, nogil, parallel, nopython):
+    def test_cython_vs_numba_frame(
+        self, sort, nogil, parallel, nopython, numba_supported_reductions
+    ):
+        func, kwargs = numba_supported_reductions
         df = DataFrame({"a": [3, 2, 3, 2], "b": range(4), "c": range(1, 5)})
         engine_kwargs = {"nogil": nogil, "parallel": parallel, "nopython": nopython}
-        result = df.groupby("a", sort=sort).mean(
-            engine="numba", engine_kwargs=engine_kwargs
+        gb = df.groupby("a", sort=sort)
+        result = getattr(gb, func)(
+            engine="numba", engine_kwargs=engine_kwargs, **kwargs
         )
-        expected = df.groupby("a", sort=sort).mean()
+        expected = getattr(gb, func)(**kwargs)
         tm.assert_frame_equal(result, expected)
 
-    def test_cython_vs_numba_getitem(self, sort, nogil, parallel, nopython):
+    def test_cython_vs_numba_getitem(
+        self, sort, nogil, parallel, nopython, numba_supported_reductions
+    ):
+        func, kwargs = numba_supported_reductions
         df = DataFrame({"a": [3, 2, 3, 2], "b": range(4), "c": range(1, 5)})
         engine_kwargs = {"nogil": nogil, "parallel": parallel, "nopython": nopython}
-        result = df.groupby("a", sort=sort)["c"].mean(
-            engine="numba", engine_kwargs=engine_kwargs
+        gb = df.groupby("a", sort=sort)["c"]
+        result = getattr(gb, func)(
+            engine="numba", engine_kwargs=engine_kwargs, **kwargs
         )
-        expected = df.groupby("a", sort=sort)["c"].mean()
+        expected = getattr(gb, func)(**kwargs)
         tm.assert_series_equal(result, expected)
 
-    def test_cython_vs_numba_series(self, sort, nogil, parallel, nopython):
+    def test_cython_vs_numba_series(
+        self, sort, nogil, parallel, nopython, numba_supported_reductions
+    ):
+        func, kwargs = numba_supported_reductions
         ser = Series(range(3), index=[1, 2, 1], name="foo")
         engine_kwargs = {"nogil": nogil, "parallel": parallel, "nopython": nopython}
-        result = ser.groupby(level=0, sort=sort).mean(
-            engine="numba", engine_kwargs=engine_kwargs
+        gb = ser.groupby(level=0, sort=sort)
+        result = getattr(gb, func)(
+            engine="numba", engine_kwargs=engine_kwargs, **kwargs
         )
-        expected = ser.groupby(level=0, sort=sort).mean()
+        expected = getattr(gb, func)(**kwargs)
         tm.assert_series_equal(result, expected)
 
-    def test_as_index_false_unsupported(self):
+    def test_as_index_false_unsupported(self, numba_supported_reductions):
+        func, kwargs = numba_supported_reductions
         df = DataFrame({"a": [3, 2, 3, 2], "b": range(4), "c": range(1, 5)})
+        gb = df.groupby("a", as_index=False)
         with pytest.raises(NotImplementedError, match="as_index=False"):
-            df.groupby("a", as_index=False).mean(engine="numba")
+            getattr(gb, func)(engine="numba", **kwargs)
 
-    def test_axis_1_unsupported(self):
+    def test_axis_1_unsupported(self, numba_supported_reductions):
+        func, kwargs = numba_supported_reductions
         df = DataFrame({"a": [3, 2, 3, 2], "b": range(4), "c": range(1, 5)})
+        gb = df.groupby("a", axis=1)
         with pytest.raises(NotImplementedError, match="axis=1"):
-            df.groupby("a", axis=1).mean(engine="numba")
+            getattr(gb, func)(engine="numba", **kwargs)