We investigate the predictable representation property (PRP) in the frame of Lévy processes. To give a general definition of the PRP we use the theory of stable subspaces. Given a Lévy process, we construct families of martingales which possess the PRP. The martingales that we consider are obtained via stochastic integration of deterministic functions with respect to the compensated Poisson random measure of the jumps of the given Lévy process.