Let X = (Xt)t≥0 be a one-dimensional L´evy process such that each Xt has a C1-density w. r. t. Lebesgue measure and certain polynomial or exponen- tial moments. We characterize all polynomially bounded functions f: R → R, and exponentially bounded functions g: R → (0, ∞), such that f (Xt) − Ef (Xt), resp. g(Xt)/Eg(Xt), are martingales.