diff --git a/aurora/batch.py b/aurora/batch.py
index eeda210..9f96356 100644
--- a/aurora/batch.py
+++ b/aurora/batch.py
@@ -42,15 +42,30 @@ class Metadata:
     rollout_step: int = 0
 
     def __post_init__(self):
-        if not torch.all(self.lat[1:] - self.lat[:-1] < 0):
-            raise ValueError("Latitudes must be strictly decreasing.")
         if not (torch.all(self.lat <= 90) and torch.all(self.lat >= -90)):
             raise ValueError("Latitudes must be in the range [-90, 90].")
-        if not torch.all(self.lon[1:] - self.lon[:-1] > 0):
-            raise ValueError("Longitudes must be strictly increasing.")
         if not (torch.all(self.lon >= 0) and torch.all(self.lon < 360)):
             raise ValueError("Longitudes must be in the range [0, 360).")
 
+        # Validate vector-valued latitudes and longitudes:
+        if self.lat.dim() == self.lon.dim() == 1:
+            if not torch.all(self.lat[1:] - self.lat[:-1] < 0):
+                raise ValueError("Latitudes must be strictly decreasing.")
+            if not torch.all(self.lon[1:] - self.lon[:-1] > 0):
+                raise ValueError("Longitudes must be strictly increasing.")
+
+        # Validate matrix-valued latitudes and longitudes:
+        elif self.lat.dim() == self.lon.dim() == 2:
+            if not torch.all(self.lat[1:, :] - self.lat[:-1, :]):
+                raise ValueError("Latitudes must be strictly decreasing along every column.")
+            if not torch.all(self.lon[:, 1:] - self.lon[:, :-1] > 0):
+                raise ValueError("Longitudes must be strictly increasing along every row.")
+
+        else:
+            raise ValueError(
+                "The latitudes and longitudes must either both be vectors or both be matrices."
+            )
+
 
 @dataclasses.dataclass
 class Batch:
diff --git a/docs/batch.md b/docs/batch.md
index 22eadba..735edb5 100644
--- a/docs/batch.md
+++ b/docs/batch.md
@@ -101,9 +101,13 @@ The following atmospheric variables are allows:
     The latitudes must be _decreasing_.
     The latitudes can either include both endpoints, like `linspace(90, -90, 721)`,
     or not include the south pole, like `linspace(90, -90, 721)[:-1]`.
+    For curvilinear grids, this can also be a matrix, in which case the foregoing conditions
+    apply to every _column_.
 * `Metadata.lon` is the vector of longitudes.
     The longitudes must be _increasing_.
     The longitudes must be in the range `[0, 360)`, so they can include zero and cannot include 360.
+    For curvilinear grids, this can also be a matrix, in which case the foregoing conditions
+    apply to every _row_.
 * `Metadata.atmos_levels` is a `tuple` of the pressure levels of the atmospheric variables in hPa.
     Note that these levels must be in exactly correspond to the order of the atmospheric variables.
     Note also that `Metadata.atmos_levels` should be a `tuple`, not a `list`.
diff --git a/tests/test_model.py b/tests/test_model.py
index 7720d2d..be006e6 100644
--- a/tests/test_model.py
+++ b/tests/test_model.py
@@ -1,19 +1,17 @@
 """Copyright (c) Microsoft Corporation. Licensed under the MIT license."""
 
 import numpy as np
+import pytest
 import torch
 
 from tests.conftest import SavedBatch
 
-from aurora import AuroraSmall, Batch
+from aurora import Aurora, AuroraSmall, Batch
 
 
-def test_aurora_small(test_input_output: tuple[Batch, SavedBatch]) -> None:
-    batch, test_output = test_input_output
-
+@pytest.fixture()
+def aurora_small() -> Aurora:
     model = AuroraSmall(use_lora=True)
-
-    # Load the checkpoint and run the model.
     model.load_checkpoint(
         "microsoft/aurora",
         "aurora-0.25-small-pretrained.ckpt",
@@ -21,8 +19,14 @@ def test_aurora_small(test_input_output: tuple[Batch, SavedBatch]) -> None:
     )
     model = model.double()
     model.eval()
+    return model
+
+
+def test_aurora_small(aurora_small: Aurora, test_input_output: tuple[Batch, SavedBatch]) -> None:
+    batch, test_output = test_input_output
+
     with torch.inference_mode():
-        pred = model.forward(batch)
+        pred = aurora_small.forward(batch)
 
     def assert_approx_equality(v_out: np.ndarray, v_ref: np.ndarray, tol: float) -> None:
         err = np.abs(v_out - v_ref).mean()
@@ -69,10 +73,50 @@ def assert_approx_equality(v_out: np.ndarray, v_ref: np.ndarray, tol: float) ->
 
 
 def test_aurora_small_decoder_init() -> None:
-    model = AuroraSmall(use_lora=True)
+    aurora_small = AuroraSmall(use_lora=True)
 
     # Check that the decoder heads are properly initialised. The biases should be zero, but the
     # weights shouldn't.
-    for layer in [*model.decoder.surf_heads.values(), *model.decoder.atmos_heads.values()]:
+    for layer in [
+        *aurora_small.decoder.surf_heads.values(),
+        *aurora_small.decoder.atmos_heads.values(),
+    ]:
         assert not torch.all(layer.weight == 0)
         assert torch.all(layer.bias == 0)
+
+
+def test_aurora_small_lat_lon_matrices(
+    aurora_small: Aurora, test_input_output: tuple[Batch, SavedBatch]
+) -> None:
+    batch, test_output = test_input_output
+
+    with torch.inference_mode():
+        pred = aurora_small.forward(batch)
+
+        # Modify the batch to have a latitude and longitude matrices.
+        n_lat = len(batch.metadata.lat)
+        n_lon = len(batch.metadata.lon)
+        batch.metadata.lat = batch.metadata.lat[:, None].expand(n_lat, n_lon)
+        batch.metadata.lon = batch.metadata.lon[None, :].expand(n_lat, n_lon)
+
+        pred_matrix = aurora_small.forward(batch)
+
+    # Check the outputs.
+    for k in pred.surf_vars:
+        np.testing.assert_allclose(
+            pred.surf_vars[k],
+            pred_matrix.surf_vars[k],
+            rtol=1e-5,
+        )
+    for k in pred.static_vars:
+        np.testing.assert_allclose(
+            pred.static_vars[k],
+            pred_matrix.static_vars[k],
+            rtol=1e-5,
+        )
+    for k in pred.atmos_vars:
+        np.testing.assert_allclose(
+            pred.atmos_vars[k],
+            pred_matrix.atmos_vars[k],
+            rtol=1e-5,
+        )