Conventional wisdom holds that quantum effects are washed out when the temperature becomes comparable to the quantized spacing between energy levels, and that the presence of a finite density of excitations above the ground state is antithetical to the stability of quantum coherent phenomena. However, the advent of modern techniques to cool and trap large collections of atoms, molecules and ions has stimulated a revival of interest in the behavior of isolated, interacting quantum systems. Recently it has been recognized that there exist "many-body localized" phases of matter that exhibit quantum dynamics even at infinite temperature, signalling a breakdown of conventional equilibrium statistical mechanics. I will discuss the intriguing and unusual properties of these inherently out-of-equilibrium systems, and outline their implications for our understanding of the phase structure of quantum many-particle systems.