  Experimental advances have allowed for the exploration of nearly isolated
quantum many-body systems whose coupling to an external bath is very weak. A
particularly interesting class of such systems is those which do not thermalize
under their own isolated quantum dynamics. In this review, we highlight the
possibility for such systems to exhibit new non-equilibrium phases of matter.
In particular, we focus on "discrete time crystals", which are many-body phases
of matter characterized by a spontaneously broken discrete time translation
symmetry. We give a definition of discrete time crystals from several points of
view, emphasizing that they are a non-equilibrium phenomenon, which is
stabilized by many-body interactions, with no analog in non-interacting
systems. We explain the theory behind several proposed models of discrete time
crystals, and compare a number of recent realizations, in different
experimental contexts.
