  Time periodic forcing in the form of coherent radiation is a standard tool
for the coherent manipulation of small quantum systems like single atoms. In
the last years, periodic driving has more and more also been considered as a
means for the coherent control of many-body systems. In particular, experiments
with ultracold quantum gases in optical lattices subjected to periodic driving
in the lower kilohertz regime have attracted a lot of attention. Milestones
include the observation of dynamic localization, the dynamic control of the
quantum phase transition between a bosonic superfluid and a Mott insulator, as
well as the dynamic creation of strong artificial magnetic fields and
topological band structures. This article reviews these recent experiments and
their theoretical description. Moreover, fundamental properties of periodically
driven many-body systems are discussed within the framework of Floquet theory,
including heating, relaxation dynamics, anomalous topological edge states, and
the response to slow parameter variations.
