Classification of Anosov diffeomorphisms is an outstanding open problem in dynamics. All currently known examples of Anosov diffeomorphisms are conjugate to nilmanifold (or infranilmanifold) Anosov automorphisms. In 1978 Farrell and Jones have constructed codimension one Anosov diffeomorphisms on higher dimensional exotic tori. We construct Anosov diffeomorphisms on manifolds that are homeomorphic to nilmanifolds yet have exotic smooth structures. These manifolds are obtained from standard nilmanifolds by connected summing with certain exotic spheres. Our construction produces Anosov diffeomorphisms of high codimension. Preliminaries on exotic structures will be thoroughly explained. This is joint work with Tom Farrell.