diff --git a/libcxx/include/__type_traits/is_nothrow_constructible.h b/libcxx/include/__type_traits/is_nothrow_constructible.h
--- a/libcxx/include/__type_traits/is_nothrow_constructible.h
+++ b/libcxx/include/__type_traits/is_nothrow_constructible.h
@@ -22,7 +22,8 @@
 
 _LIBCPP_BEGIN_NAMESPACE_STD
 
-#if __has_builtin(__is_nothrow_constructible)
+// GCC is disabled due to https://gcc.gnu.org/bugzilla/show_bug.cgi?id=106611
+#if __has_builtin(__is_nothrow_constructible) && !defined(_LIBCPP_COMPILER_GCC)
 
 template < class _Tp, class... _Args>
 struct _LIBCPP_TEMPLATE_VIS is_nothrow_constructible
diff --git a/libcxx/test/std/algorithms/robust_against_adl.compile.pass.cpp b/libcxx/test/std/algorithms/robust_against_adl.compile.pass.cpp
--- a/libcxx/test/std/algorithms/robust_against_adl.compile.pass.cpp
+++ b/libcxx/test/std/algorithms/robust_against_adl.compile.pass.cpp
@@ -11,7 +11,7 @@
 // https://buildkite.com/llvm-project/libcxx-ci/builds/15823#0184fc0b-d56b-4774-9e1d-35fe24e09e37
 // It seems like the CI gcc version is buggy. I can't reproduce the failure on my system or on
 // godbolt (https://godbolt.org/z/rsPv8e8fn).
-// UNSUPPORTED: gcc-12
+// UNSUPPORTED: gcc-12, gcc-13
 
 #include <algorithm>
 #include <cstddef>
diff --git a/libcxx/test/std/utilities/format/format.formatter/format.formatter.spec/formatter.char_array.pass.cpp b/libcxx/test/std/utilities/format/format.formatter/format.formatter.spec/formatter.char_array.pass.cpp
--- a/libcxx/test/std/utilities/format/format.formatter/format.formatter.spec/formatter.char_array.pass.cpp
+++ b/libcxx/test/std/utilities/format/format.formatter/format.formatter.spec/formatter.char_array.pass.cpp
@@ -7,7 +7,7 @@
 
 // UNSUPPORTED: c++03, c++11, c++14, c++17
 // TODO FMT __builtin_memcpy isn't constexpr in GCC
-// UNSUPPORTED: gcc-12
+// UNSUPPORTED: gcc-12, gcc-13
 
 // <format>
 
diff --git a/libcxx/test/std/utilities/meta/meta.rel/is_convertible.pass.cpp b/libcxx/test/std/utilities/meta/meta.rel/is_convertible.pass.cpp
--- a/libcxx/test/std/utilities/meta/meta.rel/is_convertible.pass.cpp
+++ b/libcxx/test/std/utilities/meta/meta.rel/is_convertible.pass.cpp
@@ -44,7 +44,6 @@
 }
 
 typedef void Function();
-typedef void ConstFunction() const;
 typedef char Array[1];
 
 struct StringType {
@@ -115,16 +114,6 @@
     test_is_not_convertible<Function*, char&> ();
     test_is_not_convertible<Function*, char*> ();
 
-    // Non-referencable function type
-    static_assert((!std::is_convertible<ConstFunction, Function>::value), "");
-    static_assert((!std::is_convertible<ConstFunction, Function*>::value), "");
-    static_assert((!std::is_convertible<ConstFunction, Function&>::value), "");
-    static_assert((!std::is_convertible<ConstFunction, Function&&>::value), "");
-    static_assert((!std::is_convertible<Function*, ConstFunction>::value), "");
-    static_assert((!std::is_convertible<Function&, ConstFunction>::value), "");
-    static_assert((!std::is_convertible<ConstFunction, ConstFunction>::value), "");
-    static_assert((!std::is_convertible<ConstFunction, void>::value), "");
-
     // Array
     test_is_not_convertible<Array, void> ();
     test_is_not_convertible<Array, Function> ();
diff --git a/libcxx/test/std/utilities/meta/meta.rel/is_convertible_fallback.pass.cpp b/libcxx/test/std/utilities/meta/meta.rel/is_convertible_fallback.pass.cpp
--- a/libcxx/test/std/utilities/meta/meta.rel/is_convertible_fallback.pass.cpp
+++ b/libcxx/test/std/utilities/meta/meta.rel/is_convertible_fallback.pass.cpp
@@ -10,6 +10,8 @@
 
 // ADDITIONAL_COMPILE_FLAGS: -D _LIBCPP_USE_IS_CONVERTIBLE_FALLBACK
 
+// UNSUPPORTED: gcc-13
+
 // type_traits
 
 // is_convertible
diff --git a/libcxx/utils/ci/buildkite-pipeline.yml b/libcxx/utils/ci/buildkite-pipeline.yml
--- a/libcxx/utils/ci/buildkite-pipeline.yml
+++ b/libcxx/utils/ci/buildkite-pipeline.yml
@@ -28,7 +28,7 @@
     # LLVM RELEASE bump remove compiler ToT - 3, e.g. "Clang 15"
     LLVM_STABLE_VERSION: "16"
     LLVM_HEAD_VERSION: "17"
-    GCC_STABLE_VERSION: "12"
+    GCC_STABLE_VERSION: "13"
 steps:
   #
   # Light pre-commit tests for things like formatting or when people forget
diff --git a/libcxx/utils/libcxx/test/params.py b/libcxx/utils/libcxx/test/params.py
--- a/libcxx/utils/libcxx/test/params.py
+++ b/libcxx/utils/libcxx/test/params.py
@@ -54,6 +54,14 @@
 
     # TODO(philnik) This fails with the PSTL.
     "-Wno-unknown-pragmas",
+
+    # TODO: Find out why GCC warns in lots of places (is this a problem with always_inline?)
+    "-Wno-dangling-reference",
+    "-Wno-mismatched-new-delete",
+    "-Wno-redundant-move",
+
+    # This doesn't make sense in real code, but we have to test it because the standard requires us to not break
+    "-Wno-self-move",
 ]
 
 _allStandards = ["c++03", "c++11", "c++14", "c++17", "c++20", "c++23", "c++26"]