A diffeomorphism f of a closed Riemannian manifold M is Anosov if TM has a splitting as a Whitney sum of two df-invariant subbundles, and df acts expansively on one of the subbundles, and contractively on the other. The only known examples of manifolds supporting an Anosov map are (certain) infranilmanifolds -- prompting Smale to ask whether manifolds having an Anosov diffeomorphism necessarily have to be infranil. In this talk, I will survey the known obstructions to having an Anosov diffeomorphism. I will also outline some recent work with Gogolev showing that products of certain aspherical manifolds with nilmanifolds do not support Anosov diffeomorphisms.