  The idea of breaking time-translation symmetry has fascinated humanity at
least since ancient proposals of the perpetuum mobile. Unlike the breaking of
other symmetries, such as spatial translation in a crystal or spin rotation in
a magnet, time translation symmetry breaking (TTSB) has been tantalisingly
elusive. We review this history up to recent developments which have shown that
discrete TTSB does takes place in periodically driven (Floquet) systems in the
presence of many-body localization. Such Floquet time-crystals represent a new
paradigm in quantum statistical mechanics --- that of an intrinsically
out-of-equilibrium many-body phase of matter.
  We include a compendium of necessary background, before specializing to a
detailed discussion of the nature, and diagnostics, of TTSB. We formalize the
notion of a time-crystal as a stable, macroscopic, conservative clock ---
explaining both the need for a many-body system in the infinite volume limit,
and for a lack of net energy absorption or dissipation.
  We also cover a range of related phenomena, including various types of
long-lived prethermal time-crystals, and expose the roles played by symmetries
-- exact and (emergent) approximate -- and their breaking. We clarify the
distinctions between many-body time-crystals and other ostensibly similar
phenomena dating as far back as the works of Faraday and Mathieu. En route, we
encounter Wilczek's suggestion that macroscopic systems should exhibit TTSB in
their ground states, together with a theorem ruling this out. We also analyze
pioneering recent experiments detecting signatures of time crystallinity in a
variety of different platforms, and provide a detailed theoretical explanation
of the physics in each case. In all existing experiments, the system does not
realize a `true' time-crystal phase, and we identify necessary ingredients for
improvements in future experiments.
