Abstract: Time dynamics of isolated many-body quantum systems has long been an elusive subject, perhaps most urgently needed in the foundations of quantum statistical mechanics. Only very recently experimentalists have begun performing detailed studies of this matter. In generic systems, one expects the nonequilibrium dynamics to lead to thermalization: a relaxation to states where the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable through the time-tested recipe of statistical mechanics. The relaxation mechanism is not obvious, however; for example, dynamical chaos cannot play the key role as it does in classical systems since quantum evolution is linear. That new rules could apply to isolated quantum systems was underscored by recent studies suggesting that statistical mechanics may give wrong predictions for their relaxation. Here we demonstrate that a generic quantum many-body system does relax to a state well-described by standard statistical mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role and that thermalization happens instead at the level of individual eigenstates, as first proposed by Srednicki [Phys. Rev. E 50, 888 (1994)]. Due to this eigenstate thermalization scenario, thermalization joins the list of processes that, like scattering, seem intrinsically temporal, and yet in quantum mechanics effectively become time-independent problems.