  Topological properties of physical systems can lead to robust behaviors that
are insensitive to microscopic details. Such topologically robust phenomena are
not limited to static systems but can also appear in driven quantum systems. In
this paper, we show that the Floquet operators of periodically driven systems
can be divided into topologically distinct (homotopy) classes, and give a
simple physical interpretation of this classification in terms of the spectra
of Floquet operators. Using this picture, we provide an intuitive understanding
of the well-known phenomenon of quantized adiabatic pumping. Systems whose
Floquet operators belong to the trivial class simulate the dynamics generated
by time-independent Hamiltonians, which can be topologically classified
according to the schemes developed for static systems. We demonstrate these
principles through an example of a periodically driven two--dimensional
hexagonal lattice model which exhibits several topological phases. Remarkably,
one of these phases supports chiral edge modes even though the bulk is
topologically trivial.
