  Closed quantum systems with quenched randomness exhibit many-body localized
regimes wherein they do not equilibrate even though prepared with macroscopic
amounts of energy above their ground states. We show that such localized
systems can order in that individual many-body eigenstates can break symmetries
or display topological order in the infinite volume limit. Indeed, isolated
localized quantum systems can order even at energy densities where the
corresponding thermally equilibrated system is disordered, i.e.: localization
protects order. In addition, localized systems can move between ordered and
disordered localized phases via non-thermodynamic transitions in the properties
of the many-body eigenstates. We give evidence that such transitions may
proceed via localized critical points. We note that localization provides
protection against decoherence that may allow experimental manipulation of
macroscopic quantum states. We also identify a `spectral transition' involving
a sharp change in the spectral statistics of the many-body Hamiltonian.
