Do quantum many-body systems necessarily come to thermal equilibrium
after a long enough time evolution? The conventional wisdom
has long been that they do and that, in the process, any quantum
information encoded in the initial state is lost irretrievably. Thus the
dynamics of many-interacting particles becomes effectively classical.
But these ingrained notions of thermalization and ergodicity have
recently been called into question. In this talk I will discuss how
ergodicity can break down in disordered quantum systems through
the phenomenon of many-body localization. In contrast to thermalizing
fluids, quantum correlations can persist through time evolution
of the localized state even at high energy densities. Thus, investigating
the many-body localization transition offers a concrete route
to address fundamental unsolved questions concerning the boundary
between classical and quantum physics in the macroscopic world. I
will emphasize the important role that quantum entanglement plays
in current attempts to understand this fascinating dynamical phase
transition. Finally I will present recent progress in confronting the
emerging theoretical understanding of many-body localization with
experimental tests using systems of ultra-cold atoms.