  We study the complexity of the classification problem for Cartan subalgebras
in von Neumann algebras. We construct a large family of II$_1$ factors whose
Cartan subalgebras up to unitary conjugacy are not classifiable by countable
structures, providing the first such examples. Additionally, we construct
examples of II$_1$ factors whose Cartan subalgebras up to conjugacy by an
automorphism are not classifiable by countable structures. Finally, we show
directly that the Cartan subalgebras of the hyperfinite II$_1$ factor up to
unitary conjugacy are not classifiable by countable structures, and deduce that
the same holds for any McDuff II$_1$ factor with at least one Cartan
subalgebra.
