diff --git a/common/map_test.cpp b/common/map_test.cpp
index 7b7cf1c9ec2f0..fcee5e0f734c1 100644
--- a/common/map_test.cpp
+++ b/common/map_test.cpp
@@ -14,11 +14,26 @@
 
 #include "common/raw_hashtable_test_helpers.h"
 
+// Workaround for std::pair comparison deficiency in libc++ 16.
+#if defined(_LIBCPP_VERSION) && _LIBCPP_VERSION < 170000
+namespace std {
+template <class T, class U, class V, class W>
+  requires(convertible_to<V, T> && convertible_to<W, U>)
+inline auto operator==(
+    pair<std::reference_wrapper<T>, std::reference_wrapper<U>> lhs,
+    pair<V, W> rhs) -> bool {
+  return lhs.first == static_cast<T>(rhs.first) &&
+         lhs.second == static_cast<U>(rhs.second);
+}
+}  // namespace std
+#endif
+
 namespace Carbon::Testing {
 namespace {
 
 using RawHashtable::FixedHashKeyContext;
 using RawHashtable::IndexKeyContext;
+using RawHashtable::MoveOnlyTestData;
 using RawHashtable::TestData;
 using RawHashtable::TestKeyContext;
 using ::testing::Pair;
@@ -29,9 +44,11 @@ auto ExpectMapElementsAre(MapT&& m, MatcherRangeT element_matchers) -> void {
   // Now collect the elements into a container.
   using KeyT = typename std::remove_reference<MapT>::type::KeyT;
   using ValueT = typename std::remove_reference<MapT>::type::ValueT;
-  std::vector<std::pair<KeyT, ValueT>> map_entries;
+  std::vector<
+      std::pair<std::reference_wrapper<KeyT>, std::reference_wrapper<ValueT>>>
+      map_entries;
   m.ForEach([&map_entries](KeyT& k, ValueT& v) {
-    map_entries.push_back({k, v});
+    map_entries.push_back({std::ref(k), std::ref(v)});
   });
 
   // Use the GoogleMock unordered container matcher to validate and show errors
@@ -68,6 +85,9 @@ auto MakeKeyValues(ValueCB value_cb, RangeT&& range, RangeTs&&... ranges)
 template <typename MapT>
 class MapTest : public ::testing::Test {};
 
+template <typename MapT>
+class MoveOnlyMapTest : public ::testing::Test {};
+
 using Types = ::testing::Types<
     Map<int, int>, Map<int, int, 16>, Map<int, int, 64>,
     Map<int, int, 0, TestKeyContext>, Map<int, int, 16, TestKeyContext>,
@@ -76,6 +96,14 @@ using Types = ::testing::Types<
     Map<TestData, TestData, 16, TestKeyContext>>;
 TYPED_TEST_SUITE(MapTest, Types);
 
+using MoveOnlyTypes = ::testing::Types<
+    Map<MoveOnlyTestData, MoveOnlyTestData>,
+    Map<MoveOnlyTestData, MoveOnlyTestData, 16>,
+    Map<MoveOnlyTestData, MoveOnlyTestData, 64>,
+    Map<MoveOnlyTestData, MoveOnlyTestData, 0, TestKeyContext>,
+    Map<MoveOnlyTestData, MoveOnlyTestData, 16, TestKeyContext>>;
+TYPED_TEST_SUITE(MoveOnlyMapTest, MoveOnlyTypes);
+
 TYPED_TEST(MapTest, Basic) {
   TypeParam m;
 
@@ -201,7 +229,7 @@ TYPED_TEST(MapTest, Move) {
   // large.
   for (int i : llvm::seq(1, 24)) {
     SCOPED_TRACE(llvm::formatv("Key: {0}", i).str());
-    ASSERT_TRUE(m.Insert(i, i * 100).is_inserted());
+    EXPECT_TRUE(m.Insert(i, i * 100).is_inserted());
   }
 
   MapT other_m1 = std::move(m);
@@ -244,10 +272,10 @@ TYPED_TEST(MapTest, Move) {
   ExpectMapElementsAre(
       other_m1, MakeKeyValues([](int k) { return k * 100; }, llvm::seq(1, 32)));
 
-  // Test copying of a moved-from table over a valid table and self-move-assign.
-  // The former is required to be valid, and the latter is in at least the case
-  // of self-move-assign-when-moved-from, but the result can be in any state so
-  // just do them and ensure we don't crash.
+  // Test copying of a moved-from table over a valid table and
+  // self-move-assign. The former is required to be valid, and the latter is
+  // in at least the case of self-move-assign-when-moved-from, but the result
+  // can be in any state so just do them and ensure we don't crash.
   MapT other_m2 = other_m1;
   // NOLINTNEXTLINE(bugprone-use-after-move): Testing required use-after-move.
   other_m2 = m;
@@ -255,6 +283,64 @@ TYPED_TEST(MapTest, Move) {
   m = std::move(m);
 }
 
+TYPED_TEST(MoveOnlyMapTest, MoveOnlyTypes) {
+  using MapT = TypeParam;
+  static_assert(!std::is_copy_assignable_v<MapT>);
+  static_assert(!std::is_copy_constructible_v<MapT>);
+  static_assert(std::is_move_assignable_v<MapT>);
+  static_assert(std::is_move_constructible_v<MapT>);
+
+  auto make_map = [] {
+    MapT m;
+    // Make sure we exceed the small size for some of the map types, but not all
+    // of them, so we cover all the combinations of moving between small and
+    // large.
+    for (int i : llvm::seq(1, 24)) {
+      SCOPED_TRACE(llvm::formatv("Key: {0}", i).str());
+      EXPECT_TRUE(m.Insert(i, i * 100).is_inserted());
+    }
+    return m;
+  };
+
+  MapT m = make_map();
+
+  MapT other_m1 = std::move(m);
+  ExpectMapElementsAre(
+      other_m1, MakeKeyValues([](int k) { return k * 100; }, llvm::seq(1, 24)));
+
+  // Add some more elements.
+  for (int i : llvm::seq(24, 32)) {
+    SCOPED_TRACE(llvm::formatv("Key: {0}", i).str());
+    ASSERT_TRUE(other_m1.Insert(i, i * 100).is_inserted());
+  }
+  ExpectMapElementsAre(
+      other_m1, MakeKeyValues([](int k) { return k * 100; }, llvm::seq(1, 32)));
+
+  // Move back over a moved-from.
+  m = std::move(other_m1);
+  ExpectMapElementsAre(
+      m, MakeKeyValues([](int k) { return k * 100; }, llvm::seq(1, 32)));
+
+  // Now add still more elements, crossing the small size limit for all tested
+  // map types.
+  for (int i : llvm::seq(32, 72)) {
+    SCOPED_TRACE(llvm::formatv("Key: {0}", i).str());
+    ASSERT_TRUE(m.Insert(i, i * 100).is_inserted());
+  }
+  ExpectMapElementsAre(
+      m, MakeKeyValues([](int k) { return k * 100; }, llvm::seq(1, 72)));
+
+  // Assignment replaces the contents.
+  m = make_map();
+  ExpectMapElementsAre(
+      m, MakeKeyValues([](int k) { return k * 100; }, llvm::seq(1, 24)));
+
+  // Self-swap (which does a self-move) works and is a no-op.
+  std::swap(m, m);
+  ExpectMapElementsAre(
+      m, MakeKeyValues([](int k) { return k * 100; }, llvm::seq(1, 24)));
+}
+
 TYPED_TEST(MapTest, Conversions) {
   using MapT = TypeParam;
   using KeyT = MapT::KeyT;
diff --git a/common/raw_hashtable.h b/common/raw_hashtable.h
index ace1e3a507994..9fc498b76cd83 100644
--- a/common/raw_hashtable.h
+++ b/common/raw_hashtable.h
@@ -163,6 +163,10 @@ struct StorageEntry {
       IsTriviallyDestructible && std::is_trivially_move_constructible_v<KeyT> &&
       std::is_trivially_move_constructible_v<ValueT>;
 
+  static constexpr bool IsCopyable =
+      IsTriviallyRelocatable || (std::is_copy_constructible_v<KeyT> &&
+                                 std::is_copy_constructible_v<ValueT>);
+
   auto key() const -> const KeyT& {
     // Ensure we don't need more alignment than available. Inside a method body
     // to apply to the complete type.
@@ -233,6 +237,9 @@ struct StorageEntry<KeyT, void> {
   static constexpr bool IsTriviallyRelocatable =
       IsTriviallyDestructible && std::is_trivially_move_constructible_v<KeyT>;
 
+  static constexpr bool IsCopyable =
+      IsTriviallyRelocatable || std::is_copy_constructible_v<KeyT>;
+
   auto key() const -> const KeyT& {
     // Ensure we don't need more alignment than available.
     static_assert(
@@ -252,7 +259,9 @@ struct StorageEntry<KeyT, void> {
     key().~KeyT();
   }
 
-  auto CopyFrom(const StorageEntry& entry) -> void {
+  auto CopyFrom(const StorageEntry& entry) -> void
+    requires(IsCopyable)
+  {
     if constexpr (IsTriviallyRelocatable) {
       memcpy(this, &entry, sizeof(StorageEntry));
     } else {
@@ -567,7 +576,8 @@ class BaseImpl {
 
   auto Construct(Storage* small_storage) -> void;
   auto Destroy() -> void;
-  auto CopySlotsFrom(const BaseImpl& arg) -> void;
+  auto CopySlotsFrom(const BaseImpl& arg) -> void
+    requires(EntryT::IsCopyable);
   auto MoveFrom(BaseImpl&& arg, Storage* small_storage) -> void;
 
   auto InsertIntoEmpty(HashCode hash) -> EntryT*;
@@ -611,9 +621,11 @@ class TableImpl : public InputBaseT {
   using BaseT = InputBaseT;
 
   TableImpl() : BaseT(SmallSize, small_storage()) {}
-  TableImpl(const TableImpl& arg);
+  TableImpl(const TableImpl& arg)
+    requires(BaseT::EntryT::IsCopyable);
   TableImpl(TableImpl&& arg) noexcept;
-  auto operator=(const TableImpl& arg) -> TableImpl&;
+  auto operator=(const TableImpl& arg) -> TableImpl&
+    requires(BaseT::EntryT::IsCopyable);
   auto operator=(TableImpl&& arg) noexcept -> TableImpl&;
   ~TableImpl();
 
@@ -1202,7 +1214,9 @@ auto BaseImpl<InputKeyT, InputValueT, InputKeyContextT>::Destroy() -> void {
 // storage.
 template <typename InputKeyT, typename InputValueT, typename InputKeyContextT>
 auto BaseImpl<InputKeyT, InputValueT, InputKeyContextT>::CopySlotsFrom(
-    const BaseImpl& arg) -> void {
+    const BaseImpl& arg) -> void
+  requires(EntryT::IsCopyable)
+{
   CARBON_DCHECK(alloc_size() == arg.alloc_size());
   ssize_t local_size = alloc_size();
 
@@ -1545,6 +1559,7 @@ BaseImpl<InputKeyT, InputValueT, InputKeyContextT>::GrowAndInsert(
 
 template <typename InputBaseT, ssize_t SmallSize>
 TableImpl<InputBaseT, SmallSize>::TableImpl(const TableImpl& arg)
+  requires(BaseT::EntryT::IsCopyable)
     : BaseT(arg.alloc_size(), arg.growth_budget_, SmallSize) {
   // Check for completely broken objects. These invariants should be true even
   // in a moved-from state.
@@ -1561,7 +1576,9 @@ TableImpl<InputBaseT, SmallSize>::TableImpl(const TableImpl& arg)
 
 template <typename InputBaseT, ssize_t SmallSize>
 auto TableImpl<InputBaseT, SmallSize>::operator=(const TableImpl& arg)
-    -> TableImpl& {
+    -> TableImpl&
+  requires(BaseT::EntryT::IsCopyable)
+{
   // Check for completely broken objects. These invariants should be true even
   // in a moved-from state.
   CARBON_DCHECK(arg.alloc_size() == 0 || !arg.is_small() ||
diff --git a/common/raw_hashtable_test_helpers.h b/common/raw_hashtable_test_helpers.h
index 83d8bf58e3abc..ee0ad5d479a91 100644
--- a/common/raw_hashtable_test_helpers.h
+++ b/common/raw_hashtable_test_helpers.h
@@ -43,6 +43,48 @@ struct TestData : Printable<TestData> {
   }
 };
 
+static_assert(std::is_copy_constructible_v<TestData>);
+
+// Non-trivial type for testing.
+struct MoveOnlyTestData : Printable<TestData> {
+  int value;
+
+  // NOLINTNEXTLINE: google-explicit-constructor
+  MoveOnlyTestData(int v) : value(v) { CARBON_CHECK(value >= 0); }
+  ~MoveOnlyTestData() {
+    CARBON_CHECK(value >= 0);
+    value = -1;
+  }
+  MoveOnlyTestData(MoveOnlyTestData&& other) noexcept
+      : MoveOnlyTestData(other.value) {
+    other.value = 0;
+  }
+  auto operator=(MoveOnlyTestData&& other) noexcept -> MoveOnlyTestData& {
+    value = other.value;
+    other.value = 0;
+    return *this;
+  }
+  auto Print(llvm::raw_ostream& out) const -> void { out << value; }
+
+  friend auto operator==(const MoveOnlyTestData& lhs,
+                         const MoveOnlyTestData& rhs) -> bool {
+    return lhs.value == rhs.value;
+  }
+
+  friend auto operator<=>(const MoveOnlyTestData& lhs,
+                          const MoveOnlyTestData& rhs) -> std::strong_ordering {
+    return lhs.value <=> rhs.value;
+  }
+
+  friend auto CarbonHashValue(const MoveOnlyTestData& data, uint64_t seed)
+      -> HashCode {
+    return Carbon::HashValue(data.value, seed);
+  }
+};
+
+static_assert(!std::is_copy_constructible_v<MoveOnlyTestData>);
+static_assert(std::is_move_constructible_v<MoveOnlyTestData>);
+
 // Test stateless key context that produces different hashes from normal.
 // Changing the hash values should result in test failures if the context ever
 // fails to be used.
diff --git a/common/set_test.cpp b/common/set_test.cpp
index 81ad810427cbb..27593025efcd8 100644
--- a/common/set_test.cpp
+++ b/common/set_test.cpp
@@ -17,6 +17,7 @@ namespace Carbon {
 namespace {
 
 using RawHashtable::IndexKeyContext;
+using RawHashtable::MoveOnlyTestData;
 using RawHashtable::TestData;
 using ::testing::UnorderedElementsAreArray;
 
@@ -24,8 +25,8 @@ template <typename SetT, typename MatcherRangeT>
 auto ExpectSetElementsAre(SetT&& s, MatcherRangeT element_matchers) -> void {
   // Collect the elements into a container.
   using KeyT = typename std::remove_reference<SetT>::type::KeyT;
-  std::vector<KeyT> entries;
-  s.ForEach([&entries](KeyT& k) { entries.push_back(k); });
+  std::vector<std::reference_wrapper<KeyT>> entries;
+  s.ForEach([&entries](KeyT& k) { entries.push_back(std::ref(k)); });
 
   // Use the GoogleMock unordered container matcher to validate and show errors
   // on wrong elements.
@@ -58,10 +59,18 @@ auto MakeElements(RangeT&& range, RangeTs&&... ranges) {
 template <typename SetT>
 class SetTest : public ::testing::Test {};
 
+template <typename SetT>
+class MoveOnlySetTest : public ::testing::Test {};
+
 using Types = ::testing::Types<Set<int>, Set<int, 16>, Set<int, 128>,
                                Set<TestData>, Set<TestData, 16>>;
 TYPED_TEST_SUITE(SetTest, Types);
 
+using MoveOnlyTypes =
+    ::testing::Types<Set<MoveOnlyTestData>, Set<MoveOnlyTestData, 16>,
+                     Set<MoveOnlyTestData, 64>>;
+TYPED_TEST_SUITE(MoveOnlySetTest, MoveOnlyTypes);
+
 TYPED_TEST(SetTest, Basic) {
   using SetT = TypeParam;
   SetT s;
@@ -162,7 +171,7 @@ TYPED_TEST(SetTest, Move) {
   // large.
   for (int i : llvm::seq(1, 24)) {
     SCOPED_TRACE(llvm::formatv("Key: {0}", i).str());
-    ASSERT_TRUE(s.Insert(i).is_inserted());
+    EXPECT_TRUE(s.Insert(i).is_inserted());
   }
 
   SetT other_s1 = std::move(s);
@@ -198,10 +207,10 @@ TYPED_TEST(SetTest, Move) {
   std::swap(other_s1, other_s1);
   ExpectSetElementsAre(other_s1, MakeElements(llvm::seq(1, 32)));
 
-  // Test copying of a moved-from table over a valid table and self-move-assign.
-  // The former is required to be valid, and the latter is in at least the case
-  // of self-move-assign-when-moved-from, but the result can be in any state so
-  // just do them and ensure we don't crash.
+  // Test copying of a moved-from table over a valid table and
+  // self-move-assign. The former is required to be valid, and the latter is
+  // in at least the case of self-move-assign-when-moved-from, but the result
+  // can be in any state so just do them and ensure we don't crash.
   SetT other_s2 = other_s1;
   // NOLINTNEXTLINE(bugprone-use-after-move): Testing required use-after-move.
   other_s2 = s;
@@ -209,6 +218,58 @@ TYPED_TEST(SetTest, Move) {
   s = std::move(s);
 }
 
+TYPED_TEST(MoveOnlySetTest, Move) {
+  using SetT = TypeParam;
+  static_assert(!std::is_copy_assignable_v<SetT>);
+  static_assert(!std::is_copy_constructible_v<SetT>);
+  static_assert(std::is_move_assignable_v<SetT>);
+  static_assert(std::is_move_constructible_v<SetT>);
+
+  auto make_set = [] {
+    SetT s;
+    // Make sure we exceed the small size for some of the set types, but not all
+    // of them, so we cover all the combinations of copying between small and
+    // large.
+    for (int i : llvm::seq(1, 24)) {
+      SCOPED_TRACE(llvm::formatv("Key: {0}", i).str());
+      EXPECT_TRUE(s.Insert(i).is_inserted());
+    }
+    return s;
+  };
+
+  SetT s = make_set();
+
+  SetT other_s1 = std::move(s);
+  ExpectSetElementsAre(other_s1, MakeElements(llvm::seq(1, 24)));
+
+  // Add some more elements.
+  for (int i : llvm::seq(24, 32)) {
+    SCOPED_TRACE(llvm::formatv("Key: {0}", i).str());
+    ASSERT_TRUE(other_s1.Insert(i).is_inserted());
+  }
+  ExpectSetElementsAre(other_s1, MakeElements(llvm::seq(1, 32)));
+
+  // Move back over a moved-from.
+  s = std::move(other_s1);
+  ExpectSetElementsAre(s, MakeElements(llvm::seq(1, 32)));
+
+  // Now add still more elements, crossing the small size limit for all tested
+  // map types.
+  for (int i : llvm::seq(32, 72)) {
+    SCOPED_TRACE(llvm::formatv("Key: {0}", i).str());
+    ASSERT_TRUE(s.Insert(i).is_inserted());
+  }
+  ExpectSetElementsAre(s, MakeElements(llvm::seq(1, 72)));
+
+  // Assignment replaces the contents.
+  s = make_set();
+  ExpectSetElementsAre(s, MakeElements(llvm::seq(1, 24)));
+
+  // Self-swap (which does a self-move) works and is a no-op.
+  std::swap(s, s);
+  ExpectSetElementsAre(s, MakeElements(llvm::seq(1, 24)));
+}
+
 TYPED_TEST(SetTest, Conversions) {
   using SetT = TypeParam;
   using KeyT = SetT::KeyT;