diff --git a/mlx/backend/cuda/CMakeLists.txt b/mlx/backend/cuda/CMakeLists.txt
index 00f639c5f1..37af7584fb 100644
--- a/mlx/backend/cuda/CMakeLists.txt
+++ b/mlx/backend/cuda/CMakeLists.txt
@@ -280,7 +280,7 @@ target_link_libraries(mlx PRIVATE CUDNN::cudnn_all)
 FetchContent_Declare(
   cutlass
   GIT_REPOSITORY https://github.com/NVIDIA/cutlass.git
-  GIT_TAG v4.3.5
+  GIT_TAG v4.4.2
   GIT_SHALLOW TRUE
   SOURCE_SUBDIR include EXCLUDE_FROM_ALL)
 FetchContent_MakeAvailable(cutlass)
diff --git a/mlx/backend/cuda/quantized/qmm/qmm.cu b/mlx/backend/cuda/quantized/qmm/qmm.cu
index a2b2784908..93982b08fd 100644
--- a/mlx/backend/cuda/quantized/qmm/qmm.cu
+++ b/mlx/backend/cuda/quantized/qmm/qmm.cu
@@ -7,6 +7,15 @@
 
 namespace mlx::core {
 
+namespace {
+
+inline bool is_last_2_dims_row_contiguous(const array& x) {
+  return x.flags().contiguous && (x.ndim() >= 2) && (x.strides(-1) == 1) &&
+      (x.strides(-2) == x.shape(-1));
+}
+
+} // namespace
+
 #if defined(MLX_CUDA_SM90A_ENABLED)
 // Defined in qmm_impl_sm90_xxx.cu files.
 template <typename TileShape, typename ClusterShape>
@@ -43,8 +52,9 @@ bool supports_qmm_sm90(
   if (!biases) {
     return false;
   }
-  if (!x.flags().row_contiguous || !w.flags().row_contiguous ||
-      !scales.flags().row_contiguous || !biases->flags().row_contiguous) {
+  if (!x.flags().row_contiguous || !is_last_2_dims_row_contiguous(w) ||
+      !is_last_2_dims_row_contiguous(scales) ||
+      !is_last_2_dims_row_contiguous(*biases)) {
     return false;
   }
   if (!transpose) {
@@ -132,11 +142,11 @@ bool supports_qmm_sm80(
   if ((n % 128 != 0) || (k % std::max(64, group_size) != 0)) {
     return false;
   }
-  if (!x.flags().row_contiguous || !w.flags().row_contiguous ||
-      !scales.flags().row_contiguous) {
+  if (!x.flags().row_contiguous || !is_last_2_dims_row_contiguous(w) ||
+      !is_last_2_dims_row_contiguous(scales)) {
     return false;
   }
-  if (biases && !biases->flags().row_contiguous) {
+  if (biases && !is_last_2_dims_row_contiguous(*biases)) {
     return false;
   }
   if (x.dtype() != float16 && x.dtype() != bfloat16) {
@@ -214,14 +224,14 @@ bool supports_qmm_naive(
     QuantizationMode mode,
     cu::Device& device) {
   int k = x.shape(-1);
-  if (k % std::max(64, group_size) != 0) {
+  if (transpose && (k % std::max(64, group_size) != 0)) {
     return false;
   }
-  if (!x.flags().row_contiguous || !w.flags().row_contiguous ||
-      !scales.flags().row_contiguous) {
+  if (!x.flags().row_contiguous || !is_last_2_dims_row_contiguous(w) ||
+      !is_last_2_dims_row_contiguous(scales)) {
     return false;
   }
-  if (biases && !biases->flags().row_contiguous) {
+  if (biases && !is_last_2_dims_row_contiguous(*biases)) {
     return false;
   }
   return true;
@@ -313,11 +323,11 @@ bool supports_qmv(
   if (k % 8 != 0) {
     return false;
   }
-  if (!x.flags().row_contiguous || !w.flags().row_contiguous ||
-      !scales.flags().row_contiguous) {
+  if (!x.flags().row_contiguous || !is_last_2_dims_row_contiguous(w) ||
+      !is_last_2_dims_row_contiguous(scales)) {
     return false;
   }
-  if (biases && !biases->flags().row_contiguous) {
+  if (biases && !is_last_2_dims_row_contiguous(*biases)) {
     return false;
   }
   if (!transpose) {
diff --git a/mlx/backend/cuda/quantized/qmm/qmm_impl_naive.cuh b/mlx/backend/cuda/quantized/qmm/qmm_impl_naive.cuh
index af52dc4d6a..9207171c66 100644
--- a/mlx/backend/cuda/quantized/qmm/qmm_impl_naive.cuh
+++ b/mlx/backend/cuda/quantized/qmm/qmm_impl_naive.cuh
@@ -40,7 +40,7 @@ cute_dequant(auto w, auto s, auto z, auto out) {
   }
 }
 
-template <typename ProblemShape, typename CtaTiler,
+template <bool HasKResidue, typename ProblemShape, typename CtaTiler,
           typename Element, typename Quant, typename Scale,
           typename StrideA, typename SmemLayoutA, typename TiledCopyA,
           typename StrideB, typename SmemLayoutB, typename TiledCopyB,
@@ -62,6 +62,20 @@ __global__ void qmm_naive_kernel(
   int thread_idx = int(threadIdx.x);
   auto [m_coord, n_coord, l_coord] = static_cast<uint3>(blockIdx);
 
+  auto m_max_coord = size<0>(shape_MNKL) - size<0>(cta_tiler) * m_coord; // M - BLK_M * m_coord
+  auto n_max_coord = size<1>(shape_MNKL) - size<1>(cta_tiler) * n_coord; // N - BLK_N * n_coord
+
+  // Shift tensor so we handle residue of K in the 0th tile.
+  auto shape_K = size<2>(shape_MNKL);
+  auto bK = size<2>(cta_tiler);
+  auto k_residue = shape_K - bK * ceil_div(shape_K, bK);
+  if constexpr (HasKResidue) {
+    A += k_residue * get<1>(dA);
+    B += k_residue * get<1>(dB) * cuda::std::min(8, sizeof_bits_v<Quant>) / 8;
+    S += k_residue * stride<1>(S_layout);
+    Z += k_residue * stride<1>(S_layout);
+  }
+
   // Represent the full tensors.
   Tensor mA_mkl = make_tensor(make_gmem_ptr(A),        select<0,2,3>(shape_MNKL), dA); // (M,K,L)
   Tensor mB_nkl = make_tensor(make_gmem_ptr<Quant>(B), select<1,2,3>(shape_MNKL), dB); // (N,K,L)
@@ -91,9 +105,6 @@ __global__ void qmm_naive_kernel(
   Tensor gS = local_tile(mS, cta_tiler, cta_coord, Step< X,_1,_1>{}); // (BLK_N,BLK_K,k)
   Tensor gZ = local_tile(mZ, cta_tiler, cta_coord, Step< X,_1,_1>{}); // (BLK_N,BLK_K,k)
 
-  auto m_max_coord = size<0>(shape_MNKL) - size<0>(gA) * m_coord; // M - BLK_M * m_coord
-  auto n_max_coord = size<1>(shape_MNKL) - size<0>(gB) * n_coord; // N - BLK_N * n_coord
-
   // Shared memory buffers.
   extern __shared__ char shared_memory[];
   using SharedStorage = SharedStorage<Element, SmemLayoutA, SmemLayoutB>;
@@ -164,17 +175,53 @@ __global__ void qmm_naive_kernel(
     __syncthreads();
   };
 
+  // Clear the rmem tiles to account for predicated off loads.
+  if constexpr (HasKResidue) {
+    clear(tArA);
+    clear(tBrB);
+    clear(tBrS);
+    clear(tBrZ);
+  }
+
   // Prefetch first tile.
-  fetch_gmem(0);
+  if constexpr (HasKResidue) {
+    Tensor tAgA_k = tAgA(_,_,_,0);
+    CUTE_UNROLL
+    for (int k = 0; k < size<2>(tArA); ++k) {
+      if (get<1>(tAcA(0,0,k)) >= -k_residue) {
+        copy_if(copy_a, tApA(_,k), tAgA_k(_,_,k), tArA(_,_,k));
+      }
+    }
+    Tensor tBgB_k = tBgB(_,_,_,0);
+    Tensor tBgS_k = tBgS(_,_,_,0);
+    Tensor tBgZ_k = tBgZ(_,_,_,0);
+    CUTE_UNROLL
+    for (int k = 0; k < size<2>(tBrB); ++k) {
+      if (get<1>(tBcB(0,0,k)) >= -k_residue) {
+        copy_if(copy_b, tBpB(_,k), tBgB_k(_,_,k), tBrB(_,_,k));
+        copy(tBgS_k(_,_,k), tBrS(_,_,k));
+        copy(tBgZ_k(_,_,k), tBrZ(_,_,k));
+      }
+    }
+  } else {
+    fetch_gmem(0);
+  }
 
   // Clear accumulators.
   clear(tCrC);
 
   // Loop over CTA tiles.
-  auto K_TILE_MAX  = size<3>(tAgA);
+  auto K_TILE_MAX = size<3>(tAgA);
   for (int tile = 0; tile < K_TILE_MAX; ++tile) {
     store_smem();
-    fetch_gmem((tile + 1 < K_TILE_MAX) ? tile + 1 : tile);
+    if constexpr (HasKResidue) {
+      // Avoid fetching full 0th-tile when there is residue.
+      if (K_TILE_MAX > 1) {
+        fetch_gmem((tile + 1 < K_TILE_MAX) ? tile + 1 : tile);
+      }
+    } else {
+      fetch_gmem((tile + 1 < K_TILE_MAX) ? tile + 1 : tile);
+    }
     gemm(mma, tCsA, tCsB, tCrC);
   }
 
@@ -236,9 +283,10 @@ inline constexpr auto make_tiled_mma() {
   }
 }
 
-template <typename T, bool KMajor = true>
+template <typename T, bool KMajor = true, bool HasKResidue = false>
 inline auto make_tiled_copy(auto num_threads, auto bM, auto bK) {
-  auto n_read = Int<8>{};
+  // TODO: Only do 1-element read for the tile of residue.
+  auto n_read = Int<HasKResidue ? 1 : 8>{};
   auto atom = Copy_Atom<UniversalCopy<uint_bit_t<n_read * sizeof_bits_v<T>>>, T>{};
   if constexpr (KMajor) {
     auto k_threads = bK / n_read;
@@ -268,7 +316,7 @@ inline constexpr auto make_scales_layout(auto n, auto k, auto l, auto group_size
   }
 }
 
-template <int TileM = 16, bool KMajor = true, bool SM80 = true,
+template <int TileM = 16, bool KMajor = true, bool SM80 = true, bool HasKResidue = false,
           typename Element, typename Quant, typename Scale>
 void qmm_naive(
     const Element* A,
@@ -314,11 +362,11 @@ void qmm_naive(
   auto sB_layout = make_smem_layout<KMajor>(bN, bK);
 
   // Atoms.
-  TiledCopy copy_a = make_tiled_copy<Element>(num_threads, bM, bK);
+  TiledCopy copy_a = make_tiled_copy<Element, true, HasKResidue>(num_threads, bM, bK);
   TiledCopy copy_b = make_tiled_copy<Quant, KMajor>(num_threads, bN, bK);
 
   auto* kernel = &qmm_naive_kernel<
-      decltype(prob_shape), decltype(cta_tiler),
+      HasKResidue, decltype(prob_shape), decltype(cta_tiler),
       Element, Quant, Scale,
       decltype(dA), decltype(sA_layout), decltype(copy_a),
       decltype(dB), decltype(sB_layout), decltype(copy_b),
@@ -348,6 +396,21 @@ void qmm_naive(
 
 namespace mlx::core {
 
+template <bool KMajor, typename F>
+inline void dispatch_k(bool has_k_residue, const char* tag, F&& f) {
+  if constexpr (KMajor) {
+    if (has_k_residue) {
+      throw std::invalid_argument(
+          fmt::format("{} K must be multiples of group_size.", tag));
+    }
+    f.template operator()<false>();
+  } else {
+    dispatch_bool(has_k_residue, [&](auto has_k_residue) {
+      f.template operator()<has_k_residue.value>();
+    });
+  }
+}
+
 template <typename F>
 inline void dispatch_element_types(Dtype dtype, const char* tag, F&& f) {
   if (dtype == float32) {
@@ -429,53 +492,55 @@ void qmm_impl_naive(
   int n = out.shape(-1);
   int k = x.shape(-1);
   int l = out.size() / (m * n);
-  bool broadcast_b = w.ndim() == 2;
+  bool broadcast_b = (w.ndim() <= 2) || (w.size() != w.data_size());
 
   bool is_sm80 = encoder.device().compute_capability_major() >= 8;
   dispatch_bool(is_sm80, [&](auto sm80) {
-    dispatch_element_types(out.dtype(), tag, [&]<typename Element>() {
-      dispatch_quant_types<Element>(
-          bits,
-          group_size,
-          mode,
-          tag,
-          [&]<typename Quant, typename Scale, int group_size>() {
-            encoder.set_input_array(x);
-            encoder.set_input_array(w);
-            encoder.set_input_array(scales);
-            if (biases) {
-              encoder.set_input_array(*biases);
-            }
-            if (lhs_indices) {
-              encoder.set_input_array(*lhs_indices);
-            }
-            if (rhs_indices) {
-              encoder.set_input_array(*rhs_indices);
-            }
-            encoder.set_output_array(out);
-            cutlass_gemm::qmm_naive<TileM, KMajor, sm80.value>(
-                gpu_ptr<Element>(x),
-                gpu_ptr<Quant>(w),
-                gpu_ptr<Scale>(scales),
-                biases ? gpu_ptr<Element>(*biases) : nullptr,
-                lhs_indices ? gpu_ptr<uint32_t>(*lhs_indices) : nullptr,
-                rhs_indices ? gpu_ptr<uint32_t>(*rhs_indices) : nullptr,
-                gpu_ptr<Element>(out),
-                m,
-                n,
-                k,
-                l,
-                broadcast_b,
-                cute::Int<group_size>{},
-                [&](auto* kernel,
-                    dim3 num_blocks,
-                    dim3 block_dims,
-                    uint32_t smem_bytes,
-                    void** args) {
-                  encoder.add_kernel_node_raw(
-                      kernel, num_blocks, block_dims, {}, smem_bytes, args);
-                });
-          });
+    dispatch_k<KMajor>(k % group_size != 0, tag, [&]<bool has_k_residue>() {
+      dispatch_element_types(out.dtype(), tag, [&]<typename Element>() {
+        dispatch_quant_types<Element>(
+            bits,
+            group_size,
+            mode,
+            tag,
+            [&]<typename Quant, typename Scale, int group_size>() {
+              encoder.set_input_array(x);
+              encoder.set_input_array(w);
+              encoder.set_input_array(scales);
+              if (biases) {
+                encoder.set_input_array(*biases);
+              }
+              if (lhs_indices) {
+                encoder.set_input_array(*lhs_indices);
+              }
+              if (rhs_indices) {
+                encoder.set_input_array(*rhs_indices);
+              }
+              encoder.set_output_array(out);
+              cutlass_gemm::qmm_naive<TileM, KMajor, sm80.value, has_k_residue>(
+                  gpu_ptr<Element>(x),
+                  gpu_ptr<Quant>(w),
+                  gpu_ptr<Scale>(scales),
+                  biases ? gpu_ptr<Element>(*biases) : nullptr,
+                  lhs_indices ? gpu_ptr<uint32_t>(*lhs_indices) : nullptr,
+                  rhs_indices ? gpu_ptr<uint32_t>(*rhs_indices) : nullptr,
+                  gpu_ptr<Element>(out),
+                  m,
+                  n,
+                  k,
+                  l,
+                  broadcast_b,
+                  cute::Int<group_size>{},
+                  [&](auto* kernel,
+                      dim3 num_blocks,
+                      dim3 block_dims,
+                      uint32_t smem_bytes,
+                      void** args) {
+                    encoder.add_kernel_node_raw(
+                        kernel, num_blocks, block_dims, {}, smem_bytes, args);
+                  });
+            });
+      });
     });
   });
 }
diff --git a/mlx/backend/cuda/quantized/qmm/qmm_impl_sm80.cuh b/mlx/backend/cuda/quantized/qmm/qmm_impl_sm80.cuh
index 62b4969e9b..302d5bd920 100644
--- a/mlx/backend/cuda/quantized/qmm/qmm_impl_sm80.cuh
+++ b/mlx/backend/cuda/quantized/qmm/qmm_impl_sm80.cuh
@@ -451,7 +451,7 @@ void qmm_impl_sm80(
   int n = out.shape(-1);
   int k = x.shape(-1);
   int l = out.size() / (m * n);
-  bool broadcast_b = w.ndim() == 2;
+  bool broadcast_b = (w.ndim() <= 2) || (w.size() != w.data_size());
 
   dispatch_element_types(out.dtype(), tag, [&]<typename Element>() {
     dispatch_quant_types<Element>(
diff --git a/mlx/backend/cuda/quantized/qmm/qmm_impl_sm90.cuh b/mlx/backend/cuda/quantized/qmm/qmm_impl_sm90.cuh
index ce79dbceeb..be552a66f1 100644
--- a/mlx/backend/cuda/quantized/qmm/qmm_impl_sm90.cuh
+++ b/mlx/backend/cuda/quantized/qmm/qmm_impl_sm90.cuh
@@ -193,7 +193,7 @@ void qmm_impl_sm90(
   int n = out.shape(-1);
   int k = x.shape(-1);
   int l = out.size() / (m * n);
-  bool broadcast_b = w.ndim() == 2;
+  bool broadcast_b = (w.ndim() <= 2) || (w.size() != w.data_size());
 
   // FIXME: Copy happens for every call.
   array scales = transpose_last_2_dims(scales_, encoder, s);
diff --git a/mlx/backend/cuda/quantized/qmm/qmv.cu b/mlx/backend/cuda/quantized/qmm/qmv.cu
index 9f70652529..540e83cd2e 100644
--- a/mlx/backend/cuda/quantized/qmm/qmv.cu
+++ b/mlx/backend/cuda/quantized/qmm/qmv.cu
@@ -406,7 +406,7 @@ void qmv(
   int n = out.shape(-1);
   int k = x.shape(-1);
   int l = out.size() / (m * n);
-  bool broadcast_w = w.ndim() == 2;
+  bool broadcast_w = (w.ndim() <= 2) || (w.size() != w.data_size());
 
   dispatch_element_types(out.dtype(), tag, [&]<typename T>() {
     dispatch_quant_types<T>(
diff --git a/python/tests/cuda_skip.py b/python/tests/cuda_skip.py
index 2de3f3bc34..31a657d713 100644
--- a/python/tests/cuda_skip.py
+++ b/python/tests/cuda_skip.py
@@ -19,11 +19,4 @@
     "TestQuantized.test_gather_qmm",
     "TestQuantized.test_gather_qmm_sorted",
     "TestQuantized.test_gather_qmm_grad",
-    "TestQuantized.test_non_multiples",
-    "TestQuantized.test_qmm_shapes",
-    "TestQuantized.test_fp_qvm",
-    "TestQuantized.test_qvm",
-    "TestQuantized.test_qmv_small_non_multiples",
-    "TestQuantized.test_small_matrix",
-    "TestExportImport.test_export_quantized_model",
 }