How a closed quantum system reaches thermal equilibrium is a fundamental question in statistical physics. Recent work has uncovered surprising richness in this process, leading to a new classification of quantum many-body systems into distinct "dynamical phases". On one end of the spectrum there are many-body localized states, which totally fail to thermalize and can retain accessible quantum correlations indefinitely. At the other end of the spectrum are maximally chaotic systems, in which quantum information gets rapidly scrambled. In this talk, I will review progress made in understanding many-body localized phases - theoretically and using experiments with ultra-cold atoms. I will then discuss a novel approach for computing the time evolution of quantum many-body systems, which captures the emergence of chaos and hydrodynamic behavior.