*Abstract:* The subject of this talk is at the crossroads of functional analysis, ergodic theory and group theory. Using a construction by Murray and von Neumann (1943), countable groups and their ergodic actions on measure spaces give rise to algebras of operators on a Hilbert space, called von Neumann algebras. A famous problem asks whether the group von Neumann algebras L(F_{n}) associated with the free groups with n generators, F_{n}, are non-isomorphic for distinct n's.While this problem is still open, its ``group measure space'' version has been settled by Sorin Popa and myself. I will comment on this, as well as on related classification results for von Neumann algebras established within Popa's deformation/rigidity program.