Localization of the eigenfunctions of quantum particles in a random potential was discovered by P.W. Anderson more than 60 years ago. In spite of its respectable maturity and intensive theoretical and experimental studies this field is far from being exhausted. Later the domain of applicability of the concept of localization was dramatically broadened. It provides an adequate framework for discussing the transition between integrable and chaotic behavior in quantum systems. The states of non-integrable many-body systems with large number of degrees of freedom is not always completely chaotic. It can be localized in the space of quantum numbers of the underlying integrable system rather than in the real space. This behavior is now known as Many-Body Localization (MBL). We will discuss several examples of the MBL transition including conventional many-body models and some matrix models exhibiting MBL. We will also discuss a connected problem of the ergodicity of the extended many-body states.