Many-body localization (MBL) generalizes Anderson localization to interacting many-body systems. MBL challenges many fundamental notions of statistical physics. In this talk I will introduce what MBL is from a math-phys perspective. Then I will discuss some striking difference between Anderson localization and MBL, in particular the existence/absence of a mobility edge, i.e. a critical energy (or energy density for many-bod physics) that separates localized states from extended states. (Joint work with W. De Roeck, M. Müller, M. Schiulaz.)