In this talk we shall discuss rigidity aspects of infinite discrete groups, which arise naturally in Geometry (as fundamental groups of manifolds), in Algebraic groups (as lattices) and, more generally, as symmetries of various mathematical objects.

Starting from classical by now rigidity results of Mostow, Margulis, Zimmer, we shall turn to the recently active area of Measurable Group Theory, which is closely related to Ergodic Theory, von Neumann algebras, and has applications to such fields as Descriptive Set Theory.