  We prove that it is not possible to classify separable von Neumann factors of
types $\II_1$, $\II_\infty$ or $\III_\lambda$, $0\leq \lambda\leq1$, up to
isomorphism by a Borel measurable assignment of "countable structures" as
invariants. In particular the isomorphism relation of type $\II_1$ factors is
not smooth. We also prove that the isomorphism relation for von Neumann $\II_1$
factors is analytic, but is not Borel.
