It is well known that every C^1 perturbation of a hyperbolic map must be conjugate to the original map via a homeomorphism (i.e. Anosov maps are structurally stable). This does not improve when one considers C^r perturbations. However, this changes when one considers several commuting hyperbolic torus automorphisms (defined by several commuting integer matrices). Here every action by several commuting smooth map which are C^1 close to the original hyperbolic automorphisms must be smoothly conjugate to the original action. This was shown first by Katok and Spatzier in the case of diagonalizable matrices. We indicate how some new ideas can be used to study the non-diagonalizable case. This is joint work with M. Einsiedler.