A function φ that is analytic and bounded in the unit disk \({\mathbb D}\) is called a generator for the Hardy space \(H^2({\mathbb D})\) or the Bergman space \(A^2({\mathbb D})\) if polynomials in φ are dense in the corresponding space. We characterize generators in terms of φ −invariant subspaces, which are also z −invariant, and study wandering properties of such subspaces. The density of bounded analytic functions in φ −invariant subspaces is also investigated.