  Dynamical phases with novel topological properties are known to arise in
driven systems of free fermions. In this paper, we obtain a `periodic table' to
describe the phases of such time-dependent systems, generalizing the periodic
table for static topological insulators. Using K-theory, we systematically
classify Floquet topological insulators from the ten Altland-Zirnbauer symmetry
classes across all dimensions. We find that the static classification scheme
described by a group $G$ becomes $G\times G$ in the time-dependent case, and
interpret the two factors as arising from the bipartite decomposition of the
unitary time-evolution operator. Topologically protected edge modes may arise
at the boundary between two Floquet systems, and we provide a mapping between
the number of such edge modes and the topological invariant of the bulk.
