We prove that it is not possible to classify separable von Neumann factors of types II, II or III, 0 ≤ λ ≤ 1, up to isomorphism by a Borel measurable assignment of "countable structures" as invariants. In particular the isomorphism relation of type II factors is not smooth. We also prove that the isomorphism relation for von Neumann II factors is analytic, but is not Borel.