  We give a systematic review of the adiabatic theorem and the leading
non-adiabatic corrections in periodically-driven (Floquet) systems. These
corrections have a two-fold origin: (i) conventional ones originating from the
gradually changing Floquet Hamiltonian and (ii) corrections originating from
changing the micro-motion operator. These corrections conspire to give a
Hall-type linear response for non-stroboscopic (time-averaged) observables
allowing one to measure the Berry curvature and the Chern number related to the
Floquet Hamiltonian, thus extending these concepts to periodically-driven
many-body systems. The non-zero Floquet Chern number allows one to realize a
Thouless energy pump, where one can adiabatically add energy to the system in
discrete units of the driving frequency. We discuss the validity of Floquet
Adiabatic Perturbation Theory (FAPT) using five different models covering
linear and non-linear few and many-particle systems. We argue that in
interacting systems, even in the stable high-frequency regimes, FAPT breaks
down at ultra slow ramp rates due to avoided crossings of photon resonances,
not captured by the inverse-frequency expansion, leading to a counter-intuitive
stronger heating at slower ramp rates. Nevertheless, large windows in the ramp
rate are shown to exist for which the physics of interacting driven systems is
well captured by FAPT.
