In the 60's D. Anosov introduced the notion of whatbecame to be called Anosov systems in his study of geodesic flows on negatively curved manifolds. Anosov diffeomorphisms and flows are the hallmark of hyperbolic and chaotic behavior. In the late 60's and early 70's itwas conjectured that Anosov diffeomorphisms have an algebraic origin, indeed that Anosov diffeomorphisms are topologically conjugated to algebraic automorphisms of infranilmanifolds. In this direction, J. Franks and A. Manning showed that if the underlying manifold is an infranilmanifold the Anosov diffeomorphism is topologically conjugated to an algebraic one. When the acting group is higher rank, strong rigidity results are expected and it is conjectured that actions of higher rank lattices with an Anosov element should be on infranilmanifolds and smoothly equivalent to algebraic. In this talk I will discuss some recent advances in this direction, proving results analogous to Franks-Manning in the setting of higher rank abelian actions and higher rank lattices on semisimple Lie groups actions.