John Franks and Michael Handel [FH2] have recently proved that any nontrivial Hamiltonian diffeomorphism of a closed surface of genus at least one has periodic orbits of arbitrarily large period. They proved a similar result for a nontrivial area-preserving diffeomorphism of a sphere with at least three fixed points. We extend these results to the case of the homeomorphisms. When the genus is at least one, we prove, moreover, that the periodic orbits may be chosen contractible if the set of contractible fixed points is contained in a disk. When the surface is a sphere, we extend the result to the case of a nontrivial homeomorphism with no wandering points. The proofs make use of an equivariant foliated version of Brouwer's plane translation theorem (see [B, Proposition 2.1]) and some properties of the linking number of fixed points