diff --git a/doc/source/whatsnew/v1.0.0.rst b/doc/source/whatsnew/v1.0.0.rst
index 29060a93923eb..d4f58ec3086b0 100644
--- a/doc/source/whatsnew/v1.0.0.rst
+++ b/doc/source/whatsnew/v1.0.0.rst
@@ -667,6 +667,7 @@ Categorical
   same type as if one used the :meth:`.str.` / :meth:`.dt.` on a :class:`Series` of that type. E.g. when accessing :meth:`Series.dt.tz_localize` on a
   :class:`Categorical` with duplicate entries, the accessor was skipping duplicates (:issue:`27952`)
 - Bug in :meth:`DataFrame.replace` and :meth:`Series.replace` that would give incorrect results on categorical data (:issue:`26988`)
+- Bug where calling :meth:`Categorical.min` or :meth:`Categorical.max` on an empty Categorical would raise a numpy exception (:issue:`30227`)
 
 
 Datetimelike
diff --git a/pandas/core/arrays/categorical.py b/pandas/core/arrays/categorical.py
index 7170aec3820a8..761d9907609c3 100644
--- a/pandas/core/arrays/categorical.py
+++ b/pandas/core/arrays/categorical.py
@@ -2116,6 +2116,10 @@ def min(self, skipna=True):
 
         Only ordered `Categoricals` have a minimum!
 
+        .. versionchanged:: 1.0.0
+
+           Returns an NA value on empty arrays
+
         Raises
         ------
         TypeError
@@ -2126,6 +2130,10 @@ def min(self, skipna=True):
         min : the minimum of this `Categorical`
         """
         self.check_for_ordered("min")
+
+        if not len(self._codes):
+            return self.dtype.na_value
+
         good = self._codes != -1
         if not good.all():
             if skipna:
@@ -2143,6 +2151,10 @@ def max(self, skipna=True):
 
         Only ordered `Categoricals` have a maximum!
 
+        .. versionchanged:: 1.0.0
+
+           Returns an NA value on empty arrays
+
         Raises
         ------
         TypeError
@@ -2153,6 +2165,10 @@ def max(self, skipna=True):
         max : the maximum of this `Categorical`
         """
         self.check_for_ordered("max")
+
+        if not len(self._codes):
+            return self.dtype.na_value
+
         good = self._codes != -1
         if not good.all():
             if skipna:
diff --git a/pandas/tests/arrays/categorical/test_analytics.py b/pandas/tests/arrays/categorical/test_analytics.py
index 637a47eba0597..3c3d9d7a44aec 100644
--- a/pandas/tests/arrays/categorical/test_analytics.py
+++ b/pandas/tests/arrays/categorical/test_analytics.py
@@ -5,22 +5,23 @@
 
 from pandas.compat import PYPY
 
-from pandas import Categorical, Index, Series
+from pandas import Categorical, Index, NaT, Series, date_range
 from pandas.api.types import is_scalar
 import pandas.util.testing as tm
 
 
 class TestCategoricalAnalytics:
-    def test_min_max(self):
-
+    @pytest.mark.parametrize("aggregation", ["min", "max"])
+    def test_min_max_not_ordered_raises(self, aggregation):
         # unordered cats have no min/max
         cat = Categorical(["a", "b", "c", "d"], ordered=False)
         msg = "Categorical is not ordered for operation {}"
-        with pytest.raises(TypeError, match=msg.format("min")):
-            cat.min()
-        with pytest.raises(TypeError, match=msg.format("max")):
-            cat.max()
+        agg_func = getattr(cat, aggregation)
+
+        with pytest.raises(TypeError, match=msg.format(aggregation)):
+            agg_func()
 
+    def test_min_max_ordered(self):
         cat = Categorical(["a", "b", "c", "d"], ordered=True)
         _min = cat.min()
         _max = cat.max()
@@ -35,6 +36,29 @@ def test_min_max(self):
         assert _min == "d"
         assert _max == "a"
 
+    @pytest.mark.parametrize(
+        "categories,expected",
+        [
+            (list("ABC"), np.NaN),
+            ([1, 2, 3], np.NaN),
+            pytest.param(
+                Series(date_range("2020-01-01", periods=3), dtype="category"),
+                NaT,
+                marks=pytest.mark.xfail(
+                    reason="https://github.com/pandas-dev/pandas/issues/29962"
+                ),
+            ),
+        ],
+    )
+    @pytest.mark.parametrize("aggregation", ["min", "max"])
+    def test_min_max_ordered_empty(self, categories, expected, aggregation):
+        # GH 30227
+        cat = Categorical([], categories=list("ABC"), ordered=True)
+
+        agg_func = getattr(cat, aggregation)
+        result = agg_func()
+        assert result is expected
+
     @pytest.mark.parametrize("skipna", [True, False])
     def test_min_max_with_nan(self, skipna):
         # GH 25303