diff --git a/Project.toml b/Project.toml
index 93db8e41..8a447f5d 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "Bijectors"
 uuid = "76274a88-744f-5084-9051-94815aaf08c4"
-version = "0.12.0"
+version = "0.12.1"
 
 [deps]
 ArgCheck = "dce04be8-c92d-5529-be00-80e4d2c0e197"
diff --git a/src/Bijectors.jl b/src/Bijectors.jl
index 9121ea70..afabd3d7 100644
--- a/src/Bijectors.jl
+++ b/src/Bijectors.jl
@@ -37,7 +37,7 @@ using LinearAlgebra: AbstractTriangular
 
 using InverseFunctions: InverseFunctions
 
-import ChangesOfVariables: with_logabsdet_jacobian
+import ChangesOfVariables: ChangesOfVariables, with_logabsdet_jacobian
 import InverseFunctions: inverse
 
 import ChainRulesCore
diff --git a/src/bijectors/pd.jl b/src/bijectors/pd.jl
index 5b57f55b..bed6ee9a 100644
--- a/src/bijectors/pd.jl
+++ b/src/bijectors/pd.jl
@@ -26,12 +26,19 @@ function logabsdetjac(b::PDBijector, X::AbstractMatrix{<:Real})
     if !issuccess(Xcf)
         Xcf = cholesky(X + max(eps(T), eps(T) * norm(X)) * I)
     end
-    return logabsdetjac(b, Xcf)
+    return logabsdetjac_pdbijector_chol(Xcf)
 end
 
-function logabsdetjac(b::PDBijector, Xcf::Cholesky)
-    U = Xcf.U
-    T = eltype(U)
-    d = size(U, 1)
-    return - sum((d .- (1:d) .+ 2) .* log.(diag(U))) - d * log(T(2))
+function logabsdetjac_pdbijector_chol(Xcf::Cholesky)
+    # NOTE: Use `UpperTriangular` here because we only need `diag(U)`
+    # and `UL` is by default already constructed in `Cholesky`.
+    UL = Xcf.UL
+    d = size(UL, 1)
+    z = sum(((d + 1):(-1):2) .* log.(diag(UL)))
+    return - (z + d * oftype(z, IrrationalConstants.logtwo))
+end
+
+# TODO: Implement explicitly.
+function with_logabsdet_jacobian(b::PDBijector, X)
+    return transform(b, X), logabsdetjac(b, X)
 end
diff --git a/src/interface.jl b/src/interface.jl
index ea10d2f0..2da09663 100644
--- a/src/interface.jl
+++ b/src/interface.jl
@@ -56,7 +56,14 @@ Broadcast.broadcastable(b::Transform) = Ref(b)
 Transform `x` using `b`, treating `x` as a single input.
 """
 transform(f::F, x) where {F<:Function} = f(x)
-transform(t::Transform, x) = first(with_logabsdet_jacobian(t, x))
+function transform(t::Transform, x)
+    res = with_logabsdet_jacobian(t, x)
+    if res isa ChangesOfVariables.NoLogAbsDetJacobian
+        error("`transform` not implemented for $(typeof(b)); implement `transform` and/or `with_logabsdet_jacobian`.")
+    end
+
+    return first(res)
+end
 
 """
     transform!(b, x[, y])
@@ -73,7 +80,14 @@ transform!(b, x, y) = copyto!(y, transform(b, x))
 
 Return `log(abs(det(J(b, x))))`, where `J(b, x)` is the jacobian of `b` at `x`.
 """
-logabsdetjac(b, x) = last(with_logabsdet_jacobian(b, x))
+function logabsdetjac(b, x)
+    res = with_logabsdet_jacobian(b, x)
+    if res isa ChangesOfVariables.NoLogAbsDetJacobian
+        error("`logabsdetjac` not implemented for $(typeof(b)); implement `logabsdetjac` and/or `with_logabsdet_jacobian`.")
+    end
+
+    return last(res)
+end
 
 """
     logabsdetjac!(b, x[, logjac])
diff --git a/test/bijectors/pd.jl b/test/bijectors/pd.jl
new file mode 100644
index 00000000..f429c5c2
--- /dev/null
+++ b/test/bijectors/pd.jl
@@ -0,0 +1,13 @@
+using Bijectors, DistributionsAD, LinearAlgebra, Test
+using Bijectors: PDBijector
+
+@testset "PDBijector" begin
+    d = 5
+    b = PDBijector()
+    dist = Wishart(d, Matrix{Float64}(I, d, d))
+    x = rand(dist)
+    # NOTE: `PDBijector` technically isn't bijective, and so the default `getjacobian`
+    # used in the ChangesOfVariables.jl tests will fail as the jacobian will have determinant 0.
+    # Hence, we disable those tests.
+    test_bijector(b, x; test_not_identity=true, changes_of_variables_test=false)
+end
diff --git a/test/bijectors/utils.jl b/test/bijectors/utils.jl
index cf1283dd..dc1d3a55 100644
--- a/test/bijectors/utils.jl
+++ b/test/bijectors/utils.jl
@@ -12,6 +12,8 @@ function test_bijector(
     logjac=nothing,
     test_not_identity=isnothing(y) && isnothing(logjac),
     test_types=false,
+    changes_of_variables_test=true,
+    inverse_functions_test=true,
     compare=isapprox,
     kwargs...
 )
@@ -29,12 +31,22 @@ function test_bijector(
     end
 
     # ChangesOfVariables.jl
-    ChangesOfVariables.test_with_logabsdet_jacobian(b, x, getjacobian; compare=compare, kwargs...)
-    ChangesOfVariables.test_with_logabsdet_jacobian(ib, isnothing(y) ? y_test : y, getjacobian; compare=compare, kwargs...)
+    # For non-bijective transformations, these tests always fail since determinant of
+    # the Jacobian is zero. Hence we allow the caller to disable them if necessary.
+    if changes_of_variables_test
+        ChangesOfVariables.test_with_logabsdet_jacobian(b, x, getjacobian; compare=compare, kwargs...)
+        ChangesOfVariables.test_with_logabsdet_jacobian(
+            ib, isnothing(y) ? y_test : y, getjacobian;
+            compare=compare,
+            kwargs...
+        )
+    end
 
     # InverseFunctions.jl
-    InverseFunctions.test_inverse(b, x; compare, kwargs...)
-    InverseFunctions.test_inverse(ib, isnothing(y) ? y_test : y; compare=compare, kwargs...)
+    if inverse_functions_test
+        InverseFunctions.test_inverse(b, x; compare, kwargs...)
+        InverseFunctions.test_inverse(ib, isnothing(y) ? y_test : y; compare=compare, kwargs...)
+    end
 
     # Always want the following to hold
     @test compare(ires[1], x; kwargs...)
diff --git a/test/runtests.jl b/test/runtests.jl
index 3185bf4c..0a00ea5f 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -35,6 +35,7 @@ if GROUP == "All" || GROUP == "Interface"
     include("bijectors/leaky_relu.jl")
     include("bijectors/coupling.jl")
     include("bijectors/ordered.jl")
+    include("bijectors/pd.jl")
 end
 
 if GROUP == "All" || GROUP == "AD"