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.