  In a periodically driven (Floquet) system, there is the possibility for new
phases of matter, not present in stationary systems, protected by discrete
time-translation symmetry. This includes topological phases protected in part
by time-translation symmetry, as well as phases distinguished by the
spontaneous breaking of this symmetry, dubbed "Floquet time crystals". We show
that such phases of matter can exist in the pre-thermal regime of
periodically-driven systems, which exists generically for sufficiently large
drive frequency, thereby eliminating the need for integrability or strong
quenched disorder that limited previous constructions. We prove a theorem that
states that such a pre-thermal regime persists until times that are nearly
exponentially-long in the ratio of certain couplings to the drive frequency. By
similar techniques, we can also construct stationary systems which
spontaneously break *continuous* time-translation symmetry. We argue
furthermore that for driven systems coupled to a cold bath, the pre-thermal
regime could potentially persist to infinite time.
