diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 3b60085e9fa66..9d0047f734c92 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -883,6 +883,7 @@ Other
 - Bug in :meth:`DataFrame.__dir__` caused a segfault when using unicode surrogates in a column name (:issue:`25509`)
 - Bug in :meth:`DataFrame.plot.scatter` caused an error when plotting variable marker sizes (:issue:`32904`)
 - :class:`IntegerArray` now implements the ``sum`` operation (:issue:`33172`)
+- More informative error message with ``np.min`` or ``np.max`` on unordered :class:`Categorical` (:issue:`33115`)
 - Bug in :class:`Tick` comparisons raising ``TypeError`` when comparing against timedelta-like objects (:issue:`34088`)
 
 .. ---------------------------------------------------------------------------
diff --git a/pandas/core/arrays/categorical.py b/pandas/core/arrays/categorical.py
index 2a01ab3802e62..80fe1ac7ce619 100644
--- a/pandas/core/arrays/categorical.py
+++ b/pandas/core/arrays/categorical.py
@@ -9,6 +9,7 @@
 
 from pandas._libs import NaT, algos as libalgos, hashtable as htable
 from pandas._typing import ArrayLike, Dtype, Ordered, Scalar
+from pandas.compat.numpy import function as nv
 from pandas.util._decorators import cache_readonly, deprecate_kwarg, doc
 from pandas.util._validators import validate_bool_kwarg, validate_fillna_kwargs
 
@@ -2077,7 +2078,7 @@ def _reduce(self, name, axis=0, **kwargs):
         return func(**kwargs)
 
     @deprecate_kwarg(old_arg_name="numeric_only", new_arg_name="skipna")
-    def min(self, skipna=True):
+    def min(self, skipna=True, **kwargs):
         """
         The minimum value of the object.
 
@@ -2096,6 +2097,7 @@ def min(self, skipna=True):
         -------
         min : the minimum of this `Categorical`
         """
+        nv.validate_min((), kwargs)
         self.check_for_ordered("min")
 
         if not len(self._codes):
@@ -2112,7 +2114,7 @@ def min(self, skipna=True):
         return self.categories[pointer]
 
     @deprecate_kwarg(old_arg_name="numeric_only", new_arg_name="skipna")
-    def max(self, skipna=True):
+    def max(self, skipna=True, **kwargs):
         """
         The maximum value of the object.
 
@@ -2131,6 +2133,7 @@ def max(self, skipna=True):
         -------
         max : the maximum of this `Categorical`
         """
+        nv.validate_max((), kwargs)
         self.check_for_ordered("max")
 
         if not len(self._codes):
diff --git a/pandas/tests/arrays/categorical/test_analytics.py b/pandas/tests/arrays/categorical/test_analytics.py
index 83ecebafbea07..4bf9b4b40d0b6 100644
--- a/pandas/tests/arrays/categorical/test_analytics.py
+++ b/pandas/tests/arrays/categorical/test_analytics.py
@@ -1,3 +1,4 @@
+import re
 import sys
 
 import numpy as np
@@ -93,6 +94,37 @@ def test_deprecate_numeric_only_min_max(self, method):
         with tm.assert_produces_warning(expected_warning=FutureWarning):
             getattr(cat, method)(numeric_only=True)
 
+    @pytest.mark.parametrize("method", ["min", "max"])
+    def test_numpy_min_max_raises(self, method):
+        cat = Categorical(["a", "b", "c", "b"], ordered=False)
+        msg = (
+            f"Categorical is not ordered for operation {method}\n"
+            "you can use .as_ordered() to change the Categorical to an ordered one"
+        )
+        method = getattr(np, method)
+        with pytest.raises(TypeError, match=re.escape(msg)):
+            method(cat)
+
+    @pytest.mark.parametrize("kwarg", ["axis", "out", "keepdims"])
+    @pytest.mark.parametrize("method", ["min", "max"])
+    def test_numpy_min_max_unsupported_kwargs_raises(self, method, kwarg):
+        cat = Categorical(["a", "b", "c", "b"], ordered=True)
+        msg = (
+            f"the '{kwarg}' parameter is not supported in the pandas implementation "
+            f"of {method}"
+        )
+        kwargs = {kwarg: 42}
+        method = getattr(np, method)
+        with pytest.raises(ValueError, match=msg):
+            method(cat, **kwargs)
+
+    @pytest.mark.parametrize("method, expected", [("min", "a"), ("max", "c")])
+    def test_numpy_min_max_axis_equals_none(self, method, expected):
+        cat = Categorical(["a", "b", "c", "b"], ordered=True)
+        method = getattr(np, method)
+        result = method(cat, axis=None)
+        assert result == expected
+
     @pytest.mark.parametrize(
         "values,categories,exp_mode",
         [