Many-body localization is a hypothesized phenomenon in which a system fails to thermalize due to random impurities. A more precise definition will be given for many-body localization. An important consequence of this definition is an area-law for the entanglement entropy of excited states with non-zero energy density. Calculations on model systems will be presented; they are consistent with the occurrence of many-body localization. Some consequences will be discussed, including the possible stabilization of topological order.