Usually the term Anderson localization is applied to the wave functions of single quantum particles in a random potential. However the concept of localization is much broader and manifests itself in various forms. It turns out that discreteness of the electronic spectrum makes the problem of quasiparticle lifetime in a finite system similar to the Anderson localization. However the localization takes place in the Fock space rather than in a real space. One can distinguish at least two different regimes: quasiparticles with high enough energies can be characterized by the finite lifetime as they are in a Fermi-liquid, while if the energy is low enough, the excitations leave infinitely long. This consideration also allows one to answer a long-standing nuclear physics question: why Wigner-Dyson statistics can be applied to the nuclear spectra?Recently it became clear [1] that the concept of the localization in the Fock space helps to attack another long-standing problem â€“ the problem of phononless hopping conductivity. The common wisdom is that even when all electronic states are localized at arbitrary low temperatures the conductivity is finite due to the phonon-induced hopping. It turns out that electron-electron interactions alone cannot cause finite conductivity at zero and low enough temperatures, however the conductivity appears at temperatures above some critical temperature Tc. From the pure theoretical point of view it means that we have a new class of finite temperature phase transitions, which are not connected with any symmetry breaking. It looks like the concept of many-body localization suggests an analytically treatable model of a glassy state and liquid-glass transition as well as of the melting of a pinned Wigner crystal. The existence of Many Body Localization justifies a revisiting of one of the basics of the statistical mechanics â€“ the postulate of the microcanonical distribution. The common wisdom is that in the presence of an arbitrary weak interaction between the particles all of the states of the system with a given energy are realized with the same probability. The crucial condition for this statement to be correct is the absence of the Many Body Localization.1. D.M. Basko, I.L. Aleiner, B.L. Altshuler, Annals of Physics 321, 1126 (2006)