  We review recent developments in the study of out-of-equilibrium topological
states of matter in isolated systems. The phenomenon of many-body localization,
exhibited by some isolated systems usually in the presence of quenched
disorder, prevents systems from equilibrating to a thermal state where the
delicate quantum correlations necessary for topological order are often washed
out. Instead, many-body localized systems can exhibit a type of eigenstate
phase structure wherein their entire many-body spectrum is characterized by
various types of quantum order, usually restricted to quantum ground states.
After introducing many-body localization and explaining how it can protect
quantum order, we then explore how the interplay of symmetry and dimensionality
with many-body localization constrains its role in stabilizing topological
phases out of equilibrium.
