Abstract: I will explain some of the recent developments in orbit equivalence theory and von Neumann algebras, including work of Gaboriau, Popa,... A tentative schedule is as follows: * Basic measure-theoretic structures (2 lectures): Measurable actions of countable groups, Orbit equivalence, crossed product von Neumann algebra, Cartan subagebras,... * Treeability, cost, L2 Betti numbers (3 or 4 lectures): Sketch of proof of Connes-Feldman-Weiss theorem. Proof of Gaboriau's non orbit equivalence of free groups theorem. L2 Betti numbers for equivalence relations following Gaboriau and/or Lück's approach. * Correspondences, Kazhdan's property T, etc. (2 lectures) * Measure-theoretic Rigidity (the remaining part of the lectures): Popa's L2 Betti numbers paper Proof of Popa's cocycle superridigity theorem Parts of Popa's von Neumann rigidity theorems (i.e. the "malleable" papers) Parts of Ozawa-Popa's recent preprint (unicity of Cartan subalgebras) Precise references will be given during the lectures.