  The question whether Anderson insulators can persist to finite-strength
interactions - a scenario dubbed many-body localization - has recently received
a great deal of interest. The origin of such a many-body localized phase has
been described as localization in Fock space, a picture we examine numerically.
We then formulate a precise sense in which a single energy eigenstate of a
Hamiltonian can be adiabatically connected to a state of a non-interacting
Anderson insulator. We call such a state a many-body localized state and define
a many-body localized phase as one in which almost all states are many-body
localized states. We explore the possible consequences of this; the most
striking is an area law for the entanglement entropy of almost all excited
states in a many-body localized phase. We present the results of numerical
calculations for a one-dimensional system of spinless fermions. Our results are
consistent with an area law and, by implication, many-body localization for
almost all states and almost all regions for weak enough interactions and
strong disorder. However, there are rare regions and rare states with much
larger entanglement entropies. Furthermore, we study the implications that
many-body localization may have for topological phases and self-correcting
quantum memories. We find that there are scenarios in which many-body
localization can help to stabilize topological order at non-zero energy
density, and we propose potentially useful criteria to confirm these scenarios.
