diff --git a/.buildkite/pipeline.yml b/.buildkite/pipeline.yml
index 43999a1df..015aebeb0 100644
--- a/.buildkite/pipeline.yml
+++ b/.buildkite/pipeline.yml
@@ -5,6 +5,8 @@ steps:
         version:
           - "1.10"
           - "1.11"
+          - "1.12"
+          - "1.13"
     plugins:
       - JuliaCI/julia#v1:
           version: "{{matrix.version}}"
@@ -14,8 +16,7 @@ steps:
       julia -e 'println("--- :julia: Instantiating project")
                 using Pkg
                 Pkg.develop(; path=pwd())
-                Pkg.develop(; name="CUDA")' || exit 3
-
+                Pkg.add(url="https://github.com/eschnett/CUDA.jl", rev="eschnett/julia-1.13")' || exit 3
       julia -e 'println("+++ :julia: Running tests")
                 using Pkg
                 Pkg.test("CUDA"; coverage=true, test_args=["base/kernelabstractions"])'
diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index f80d0651b..c7df8a00f 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -27,7 +27,9 @@ jobs:
           - '1.8'
           - '1.9'
           - '1.10'
-          - '~1.11.0-0'
+          - '1.11'
+          - '1.12'
+          - '1.13-nightly'
         os:
           - ubuntu-latest
           - macOS-latest
diff --git a/test/compiler.jl b/test/compiler.jl
index 85312262e..fac078ab8 100644
--- a/test/compiler.jl
+++ b/test/compiler.jl
@@ -24,11 +24,19 @@ end
     A[1] = Base.Checked.checked_add(a, b)
 end
 
+@static if VERSION > v"1.12"
+    const MethodOrCodeInstance = Core.CodeInstance
+    _getname(mi) = mi.def.def.name
+else
+    const MethodOrCodeInstance = Core.MethodInstance
+    _getname(mi) = mi.def.name
+end
+
 function check_for_overdub(stmt)
     if stmt isa Expr
         if stmt.head == :invoke
-            mi = first(stmt.args)::Core.MethodInstance
-            if mi.def.name === :overdub
+            mi = first(stmt.args)::MethodOrCodeInstance
+            if _getname(mi) === :overdub
                 @show stmt
                 return true
             end
diff --git a/test/runtests.jl b/test/runtests.jl
index bf9b445a5..f395b70dc 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -36,7 +36,7 @@ if Base.JLOptions().check_bounds == 0 || Base.JLOptions().check_bounds == 1
 end
 
 if Base.JLOptions().check_bounds == 0 || Base.JLOptions().check_bounds == 2
-    @kernel inbounds = true function my_bounded_kernel(a)
+    @kernel inbounds = true function my_inbounds_kernel(a)
         idx = @index(Global, Linear)
         a[idx] = 0
     end
@@ -73,7 +73,8 @@ struct NewBackend <: KernelAbstractions.GPU end
 end
 
 include("extensions/enzyme.jl")
-@static if VERSION >= v"1.7.0"
+# The Enzyme tests fail with Julia 1.13. They also fail with Julia 1.12 on Windows.
+@static if VERSION >= v"1.7.0" && VERSION < v"1.13-" && !(Sys.iswindows() && VERSION >= v"1.12-")
     @testset "Enzyme" begin
         enzyme_testsuite(CPU, Array)
     end
diff --git a/test/test.jl b/test/test.jl
index 8e3bd7136..d0d3e1d07 100644
--- a/test/test.jl
+++ b/test/test.jl
@@ -206,7 +206,11 @@ function unittest_testsuite(Backend, backend_str, backend_mod, BackendArrayT; sk
                 @test occursin("!alias.scope", IR)
                 @test occursin("!noalias", IR)
             elseif backend_str == "CUDA"
-                @test occursin("@llvm.nvvm.ldg", IR)
+                if Base.libllvm_version >= v"20"
+                    @test occursin("addrspace(1)", IR)
+                else
+                    @test occursin("@llvm.nvvm.ldg", IR)
+                end
             elseif backend_str == "ROCM"
                 @test occursin("addrspace(4)", IR)
             else