What happens in an isolated quantum system when both disorder and
interactions are present? Over the recent years, the picture of a
non-thermalizing phase of matter, the many-localized phase, has emerged as a
stable solution. We present a basic introduction to the topic of many-body
localization, using the simple example of a quantum spin chain which allows us
to illustrate several of the properties of this phase. We then briefly review
the current experimental research efforts probing this physics. The largest
part of this review is a selection of more specialized questions, some of which
are currently under active investigation. We conclude by summarizing the
connections between many-body localization and quantum simulations.Comment: Review article. 28 pages, 8 figures, Comptes Rendus Physique (2018
