diff --git a/doc/source/whatsnew/v1.3.0.rst b/doc/source/whatsnew/v1.3.0.rst
index 7189c6e68d53d..6f169b2f73242 100644
--- a/doc/source/whatsnew/v1.3.0.rst
+++ b/doc/source/whatsnew/v1.3.0.rst
@@ -877,7 +877,7 @@ Sparse
 
 - Bug in :meth:`DataFrame.sparse.to_coo` raising ``KeyError`` with columns that are a numeric :class:`Index` without a 0 (:issue:`18414`)
 - Bug in :meth:`SparseArray.astype` with ``copy=False`` producing incorrect results when going from integer dtype to floating dtype (:issue:`34456`)
--
+- Implemented :meth:`SparseArray.max` and :meth:`SparseArray.min` (:issue:`40921`)
 
 ExtensionArray
 ^^^^^^^^^^^^^^
diff --git a/pandas/core/arrays/sparse/array.py b/pandas/core/arrays/sparse/array.py
index 37898ce682e4f..091efa68c67da 100644
--- a/pandas/core/arrays/sparse/array.py
+++ b/pandas/core/arrays/sparse/array.py
@@ -1392,6 +1392,24 @@ def mean(self, axis=0, *args, **kwargs):
             nsparse = self.sp_index.ngaps
             return (sp_sum + self.fill_value * nsparse) / (ct + nsparse)
 
+    def max(self, axis=0, *args, **kwargs):
+        nv.validate_max(args, kwargs)
+
+        # This condition returns a nan if there are no valid values in the array.
+        if self.size > 0 and self._valid_sp_values.size == 0:
+            return np.nan
+        else:
+            return np.nanmax(self, axis)
+
+    def min(self, axis=0, *args, **kwargs):
+        nv.validate_min(args, kwargs)
+
+        # This condition returns a nan if there are no valid values in the array.
+        if self.size > 0 and self._valid_sp_values.size == 0:
+            return np.nan
+        else:
+            return np.nanmin(self, axis)
+
     # ------------------------------------------------------------------------
     # Ufuncs
     # ------------------------------------------------------------------------
diff --git a/pandas/tests/arrays/sparse/test_array.py b/pandas/tests/arrays/sparse/test_array.py
index e073871f96bb4..a96e5b07b7f7e 100644
--- a/pandas/tests/arrays/sparse/test_array.py
+++ b/pandas/tests/arrays/sparse/test_array.py
@@ -1311,3 +1311,25 @@ def test_dropna(fill_value):
     df = pd.DataFrame({"a": [0, 1], "b": arr})
     expected_df = pd.DataFrame({"a": [1], "b": exp}, index=pd.Int64Index([1]))
     tm.assert_equal(df.dropna(), expected_df)
+
+
+class TestMinMax:
+    plain_data = np.arange(5).astype(float)
+    data_neg = plain_data * (-1)
+    data_NaN = SparseArray(np.array([0, 1, 2, np.nan, 4]))
+    data_all_NaN = SparseArray(np.array([np.nan, np.nan, np.nan, np.nan, np.nan]))
+
+    @pytest.mark.parametrize(
+        "raw_data,max_expected,min_expected",
+        [
+            (plain_data, [4], [0]),
+            (data_neg, [0], [-4]),
+            (data_NaN, [4], [0]),
+            (data_all_NaN, [np.nan], [np.nan]),
+        ],
+    )
+    def test_maxmin(self, raw_data, max_expected, min_expected):
+        max_result = SparseArray(raw_data).max()
+        min_result = SparseArray(raw_data).min()
+        assert max_result in max_expected
+        assert min_result in min_expected