Orbit equivalence (OE) theory has seen an explosion of activity in the last twenty years. This progress has been in part triggered by the success of Sorin Popa's deformation/rigidity approach to the classification of II_1 factors. In this talk we will survey some of these results, discuss about OE superrigid actions (i.e. actions that are completely remembered by their OE relation) and present a new product rigidity phenomenon in OE. We will also show how techniques from von Neumann algebras can be used in the classification of actions up to OE.