  We present a theory of periodically driven, many-body localized (MBL)
systems. We argue that MBL persists under periodic driving at high enough
driving frequency: The Floquet operator (evolution operator over one driving
period) can be represented as an exponential of an effective time-independent
Hamiltonian, which is a sum of quasi-local terms and is itself fully MBL. We
derive this result by constructing a sequence of canonical transformations to
remove the time-dependence from the original Hamiltonian. When the driving
evolves smoothly in time, the theory can be sharpened by estimating the
probability of adiabatic Landau-Zener transitions at many-body level crossings.
In all cases, we argue that there is delocalization at sufficiently low
frequency. We propose a phase diagram of driven MBL systems.
