diff --git a/haiku/_src/pool.py b/haiku/_src/pool.py
index be5e3d8d4..1fa61199e 100644
--- a/haiku/_src/pool.py
+++ b/haiku/_src/pool.py
@@ -129,6 +129,10 @@ def avg_pool(
     raise ValueError(f"Invalid padding '{padding}', must be 'SAME' or 'VALID'.")
 
   value = jnp.asarray(value)
+  value_dtype = value.dtype
+  if value_dtype in (jnp.float16, jnp.bfloat16):
+    # Accumulate small floating-point dtypes in float32 before averaging.
+    value = value.astype(jnp.float32)
   _warn_if_unsafe(window_shape, strides)
   window_shape = _infer_shape(value, window_shape, channel_axis)
   strides = _infer_shape(value, strides, channel_axis)
@@ -137,7 +141,7 @@ def avg_pool(
   pooled = lax.reduce_window(value, *reduce_window_args)
   if padding == "VALID":
     # Avoid the extra reduce_window.
-    return pooled / np.prod(window_shape)
+    return (pooled / np.prod(window_shape)).astype(value_dtype)
   else:
     # Count the number of valid entries at each input point, then use that for
     # computing average. Assumes that any two arrays of same shape will be
@@ -145,7 +149,7 @@ def avg_pool(
     shape = [(v if w != 1 else 1) for (v, w) in zip(value.shape, window_shape)]
     window_counts = lax.reduce_window(
         jnp.ones(shape, value.dtype), *reduce_window_args)
-    return pooled / window_counts
+    return (pooled / window_counts).astype(value_dtype)
 
 
 class MaxPool(hk.Module):
diff --git a/haiku/_src/pool_precision_test.py b/haiku/_src/pool_precision_test.py
new file mode 100644
index 000000000..daa11186c
--- /dev/null
+++ b/haiku/_src/pool_precision_test.py
@@ -0,0 +1,107 @@
+# Copyright 2019 DeepMind Technologies Limited. All Rights Reserved.
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+# ==============================================================================
+"""Low-precision average pooling accumulates before narrowing its result."""
+
+import itertools
+from absl.testing import absltest
+from absl.testing import parameterized
+import haiku as hk
+import jax
+import jax.numpy as jnp
+import numpy as np
+
+
+def numpy_pool(x, window, strides, padding):
+  x = np.asarray(x, dtype=np.float64)
+  out_shape = []
+  pads = []
+  for n, w, s in zip(x.shape, window, strides):
+    size = (n + s - 1) // s if padding == "SAME" else max((n - w) // s + 1, 0)
+    total = max((size - 1) * s + w - n, 0) if padding == "SAME" else 0
+    out_shape.append(size)
+    pads.append(total // 2)
+  out = np.empty(out_shape)
+  for index in itertools.product(*(range(n) for n in out_shape)):
+    slices = []
+    for i, n, w, s, p in zip(index, x.shape, window, strides, pads):
+      start = i * s - p
+      slices.append(slice(max(start, 0), min(start + w, n)))
+    out[index] = x[tuple(slices)].mean()
+  return out
+
+
+class PoolPrecisionTest(parameterized.TestCase):
+
+  @parameterized.product(
+      padding=["SAME", "VALID"],
+      channels_first=[False, True],
+      value=[256.0, -256.0],
+  )
+  def test_representable_constant_averages_do_not_overflow(
+      self, padding, channels_first, value
+  ):
+    shape = (1, 2, 16, 16) if channels_first else (1, 16, 16, 2)
+    window = (1, 1, 16, 16) if channels_first else (1, 16, 16, 1)
+    strides = (1, 1, 8, 8) if channels_first else (1, 8, 8, 1)
+    x = jnp.full(shape, value, jnp.float16)
+    expected = numpy_pool(x, window, strides, padding)
+    fn = lambda inputs: hk.avg_pool(inputs, window, strides, padding)
+    for apply in (fn, jax.jit(fn)):
+      actual = apply(x)
+      self.assertEqual(actual.dtype, jnp.float16)
+      np.testing.assert_array_equal(actual, expected)
+
+  @parameterized.product(
+      dtype=[jnp.float16, jnp.bfloat16, jnp.float32], padding=["SAME", "VALID"]
+  )
+  def test_nonuniform_windows_match_high_precision_reference(
+      self, dtype, padding
+  ):
+    values = (np.arange(256).reshape(1, 16, 16, 1) % 13 - 6) * 100 + 400
+    x = jnp.asarray(values, dtype)
+    window = (1, 8, 8, 1)
+    strides = (1, 5, 5, 1)
+    expected = jnp.asarray(numpy_pool(x, window, strides, padding), dtype)
+    actual = hk.avg_pool(x, window, strides, padding)
+    self.assertEqual(actual.dtype, dtype)
+    np.testing.assert_allclose(
+        actual, expected, rtol=0.008 if dtype == jnp.bfloat16 else 0.001, atol=0
+    )
+
+  def test_avgpool_module_has_finite_loss_and_parameter_gradients(self):
+    transformed = hk.without_apply_rng(
+        hk.transform(lambda x: hk.AvgPool((1, 16, 16, 1), 1, "VALID")(x))
+    )
+    x = jnp.full((1, 16, 16, 1), 256.0, jnp.float16)
+    params = transformed.init(jax.random.PRNGKey(0), x)
+    objective = lambda z: jnp.square(
+        transformed.apply(params, z).astype(jnp.float32) - 100
+    ).sum()
+    loss, grad = jax.jit(jax.value_and_grad(objective))(x)
+    np.testing.assert_allclose(loss, 156.0**2)
+    np.testing.assert_array_equal(grad, np.full(x.shape, 2 * 156 / 256))
+
+  def test_batched_and_vmapped_pooling_agree(self):
+    x = jnp.full((2, 16, 16, 1), 256.0, jnp.float16)
+    individual = lambda image: hk.avg_pool(
+        image, (16, 16, 1), (8, 8, 1), "SAME"
+    )
+    batched = hk.avg_pool(x, (16, 16, 1), (8, 8, 1), "SAME")
+    np.testing.assert_array_equal(jax.jit(jax.vmap(individual))(x), batched)
+    self.assertTrue(bool(jnp.isfinite(batched).all()))
+
+
+if __name__ == "__main__":
+  absltest.main()