We study the local regularity of solutions f to the integro-differential equationAf=ginU for open sets U⊆ ℝ^d, where A is the infinitesimal generator of a Lévy process (X_t)_{t≥ 0}. Under the assumption that the transition density of (X_t)_{t≥ 0} satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions f. Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.