Usually the term Anderson localization applies to wave functions of a single quantum particle in a random potential. However the concept of localization is much broader. It turns out that discreteness of the electronic spectrum makes the problem of quasiparticle lifetime in a finite system similar to Anderson localization. However the localization takes place in Fock space rather than in real space. This answers a long-standing nuclear physics question: why can Wigner-Dyson statistics be applied to nuclear spectra? Similarly, the problem of phononless hopping conductivity can be attacked. It turns out that electron-electron interactions alone cannot cause finite conductivity at low enough temperatures, but the conductivity appears above some critical temperature Tc. This is a new class of finite temperature phase transitions, not connected with symmetry breaking. 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.