  We consider Mott insulators driven by periodic coherent laser radiation,
using both single orbital and multi-orbital models, noting that the latter is
of more interest in solid state systems. We derive general expressions for the
resulting periodically driven spin models and spin-orbital models using
time-dependent perturbation theory. First, we show that the effective exchange
interactions of the Floquet Hamiltonians are highly tunable by the frequency,
amplitude, and polarization of the laser. Second, we take the effect of finite
bandwidth of excitations into account and study possible heating effects. Using
the same formalism with a slight modification we also consider the small
frequency regime and study the dielectric breakdown of Mott insulators.
