diff --git a/doc/release/upcoming_changes/28343.change.rst b/doc/release/upcoming_changes/28343.change.rst
new file mode 100644
index 000000000000..378ef775b62e
--- /dev/null
+++ b/doc/release/upcoming_changes/28343.change.rst
@@ -0,0 +1 @@
+* The vector norm ``ord=inf`` and the matrix norms ``ord={1, 2, inf, 'nuc'}`` now always returns zero for empty arrays. Empty arrays have at least one axis of size zero. This affects `np.linalg.norm`, `np.linalg.vector_norm`, and `np.linalg.matrix_norm`. Previously, NumPy would raises errors or return zero depending on the shape of the array.
diff --git a/numpy/linalg/_linalg.py b/numpy/linalg/_linalg.py
index 32a52b7b8cdb..f5bc524a27ae 100644
--- a/numpy/linalg/_linalg.py
+++ b/numpy/linalg/_linalg.py
@@ -2541,7 +2541,7 @@ def lstsq(a, b, rcond=None):
     return wrap(x), wrap(resids), rank, s
 
 
-def _multi_svd_norm(x, row_axis, col_axis, op):
+def _multi_svd_norm(x, row_axis, col_axis, op, initial=None):
     """Compute a function of the singular values of the 2-D matrices in `x`.
 
     This is a private utility function used by `numpy.linalg.norm()`.
@@ -2565,7 +2565,7 @@ def _multi_svd_norm(x, row_axis, col_axis, op):
 
     """
     y = moveaxis(x, (row_axis, col_axis), (-2, -1))
-    result = op(svd(y, compute_uv=False), axis=-1)
+    result = op(svd(y, compute_uv=False), axis=-1, initial=initial)
     return result
 
 
@@ -2763,7 +2763,7 @@ def norm(x, ord=None, axis=None, keepdims=False):
 
     if len(axis) == 1:
         if ord == inf:
-            return abs(x).max(axis=axis, keepdims=keepdims)
+            return abs(x).max(axis=axis, keepdims=keepdims, initial=0)
         elif ord == -inf:
             return abs(x).min(axis=axis, keepdims=keepdims)
         elif ord == 0:
@@ -2797,17 +2797,17 @@ def norm(x, ord=None, axis=None, keepdims=False):
         if row_axis == col_axis:
             raise ValueError('Duplicate axes given.')
         if ord == 2:
-            ret = _multi_svd_norm(x, row_axis, col_axis, amax)
+            ret = _multi_svd_norm(x, row_axis, col_axis, amax, 0)
         elif ord == -2:
             ret = _multi_svd_norm(x, row_axis, col_axis, amin)
         elif ord == 1:
             if col_axis > row_axis:
                 col_axis -= 1
-            ret = add.reduce(abs(x), axis=row_axis).max(axis=col_axis)
+            ret = add.reduce(abs(x), axis=row_axis).max(axis=col_axis, initial=0)
         elif ord == inf:
             if row_axis > col_axis:
                 row_axis -= 1
-            ret = add.reduce(abs(x), axis=col_axis).max(axis=row_axis)
+            ret = add.reduce(abs(x), axis=col_axis).max(axis=row_axis, initial=0)
         elif ord == -1:
             if col_axis > row_axis:
                 col_axis -= 1
@@ -2819,7 +2819,7 @@ def norm(x, ord=None, axis=None, keepdims=False):
         elif ord in [None, 'fro', 'f']:
             ret = sqrt(add.reduce((x.conj() * x).real, axis=axis))
         elif ord == 'nuc':
-            ret = _multi_svd_norm(x, row_axis, col_axis, sum)
+            ret = _multi_svd_norm(x, row_axis, col_axis, sum, 0)
         else:
             raise ValueError("Invalid norm order for matrices.")
         if keepdims:
diff --git a/numpy/linalg/tests/test_linalg.py b/numpy/linalg/tests/test_linalg.py
index 356676e7ce6d..1a79629814e9 100644
--- a/numpy/linalg/tests/test_linalg.py
+++ b/numpy/linalg/tests/test_linalg.py
@@ -2372,6 +2372,16 @@ def test_matrix_norm():
     assert_almost_equal(actual, np.array([[14.2828]]), double_decimal=3)
 
 
+def test_matrix_norm_empty():
+    for shape in [(0, 2), (2, 0), (0, 0)]:
+        for dtype in [np.float64, np.float32, np.int32]:
+            x = np.zeros(shape, dtype)
+            assert_equal(np.linalg.matrix_norm(x, ord="fro"), 0)
+            assert_equal(np.linalg.matrix_norm(x, ord="nuc"), 0)
+            assert_equal(np.linalg.matrix_norm(x, ord=1), 0)
+            assert_equal(np.linalg.matrix_norm(x, ord=2), 0)
+            assert_equal(np.linalg.matrix_norm(x, ord=np.inf), 0)
+
 def test_vector_norm():
     x = np.arange(9).reshape((3, 3))
     actual = np.linalg.vector_norm(x)
@@ -2388,3 +2398,11 @@ def test_vector_norm():
     expected = np.full((1, 1), 14.2828, dtype='float64')
     assert_equal(actual.shape, expected.shape)
     assert_almost_equal(actual, expected, double_decimal=3)
+
+
+def test_vector_norm_empty():
+    for dtype in [np.float64, np.float32, np.int32]:
+        x = np.zeros(0, dtype)
+        assert_equal(np.linalg.vector_norm(x, ord=1), 0)
+        assert_equal(np.linalg.vector_norm(x, ord=2), 0)
+        assert_equal(np.linalg.vector_norm(x, ord=np.inf), 0)