Actions of groups Zk and Rk for k>1 in many cases exhibit a behaviour very different from their one-dimensional counterparts (diffeomorphisms and flows), providing that they do not reduce to Z or R actions. In particular, invariant measures seem to be scarse, cocycles over such actions tend to be trivial and differentiable orbit structure is preserved under small perturbations. Those phenomena are known as measure, cocycle and differentiable rigidity.

I will give a brief overview of some results and conjectures concerning rigidity of higher rank abelian actions and give a short outline of the proof of cocycle and local differentiable rigidity for Zk actions by ergodic toral automorphisms.