  We give a general overview of the high-frequency regime in periodically
driven systems and identify three distinct classes of driving protocols in
which the infinite-frequency Floquet Hamiltonian is not equal to the
time-averaged Hamiltonian. These classes cover systems, such as the Kapitza
pendulum, the Harper-Hofstadter model of neutral atoms in a magnetic field, the
Haldane Floquet Chern insulator and others. In all setups considered, we
discuss both the infinite-frequency limit and the leading finite-frequency
corrections to the Floquet Hamiltonian. We provide a short overview of Floquet
theory focusing on the gauge structure associated with the choice of
stroboscopic frame and the differences between stroboscopic and
non-stroboscopic dynamics. In the latter case one has to work with dressed
operators representing observables and a dressed density matrix. We also
comment on the application of Floquet Theory to systems described by static
Hamiltonians with well-separated energy scales and, in particular, discuss
parallels between the inverse-frequency expansion and the Schrieffer-Wolff
transformation extending the latter to driven systems.

Quantum Gases;Statistical Mechanics