Many Body Localization

Famously, Anderson showed in 1958 that non-interacting quantum mechanical electrons placed in a random potential could fail to exhibit diffusion. In 2005 Basko, Aleiner and Altshuler put the long conjectured extension of the Anderson localization to the interacting setting, termed Many Body Localization, on a much firmer footing setting off a rapidly growing volume of work on this phenomenon. I will describe the broad outlines of this work and discuss how it has both enriched our understanding of the foundation of quantum statistical mechanics and led to the remarkable notion of non-equilibrium quantum phase structure that lies outside the former.