In 1932 von Neumann proposed classifying the statistical behavior of diffeomorphisms of manifolds. In modern language this is restated as classifying the equivalence relation of measure theoretic isomorphism. The main theorem in this talk is that the classification problem is impossible. The equivalence relation is not Borel. As part of the proof a “Global Structure Theorem” is proved that settle a couple of open questions about another classical problem, the “realization problem.” This is joint work with B. Weiss.