Hyperbolic flows and transformations on compact manifolds are among the most well-studied objects in modern dynamical systems. In the case of a transformation, topological rigidity phenomenon are expected. For example, every transitive hyperbolic diffeomorphism of a torus is conjugated in the homeomorphism group to an automorphism. However, it is rare that the conjugating homeomorphism is differentiable. For flows, one obtains homeomorphic orbit foliations, but it is again rare that the homeomorphism is differentiable and time-preserving. I will report on recent progress in the study of hyperbolic group actions pioneered by Katok, Spatzier and Zimmer, where such differentiability and time-preservation are always expected.