  We develop a low-frequency perturbation theory in the extended Floquet
Hilbert space of a periodically driven quantum systems, which puts the high-
and low-frequency approximations to the Floquet theory on the same footing. It
captures adiabatic perturbation theories recently discussed in the literature
as well as diabatic deviation due to Floquet resonances. For illustration, we
apply our Floquet perturbation theory to a driven two-level system as in the
Schwinger-Rabi and the Landau-Zener-St\"uckelberg-Majorana models. We reproduce
some known expressions for transition probabilities in a simple and systematic
way and clarify and extend their regime of applicability. We then apply the
theory to a periodically-driven system of fermions on the lattice and obtain
the spectral properties and the low-frequency dynamics of the system.
