Periodic orbits are studied using generating functions. We develop necessary and su#cient conditions for existence of periodic orbits of a given period or going through a given point in space. These conditions reduce the search to either solving a set of implicit equations, which can often be done graphically, or finding the roots of an equation of one variable only, independent of the Hamiltonian system considered. We apply our results to periodic orbits in the vicinity of other periodic orbits and around the Libration points in the three-body problem