Self-decomposable probability measures μ on ℝ+ are characterized in terms of minus the logarithm of the Laplace transform of μ, say f, by the requirement that s→sf′(s) is again minus the logarithm of the Laplace transform of an infinitely divisible probability on ℝ+. Iteration of this condition yields characterizations in the case of ℝ+ of Urbanik's classes Ln of multiply self-decomposable probabilities. The analogous characterization for discrete (multiply) self-decomposable probabilities on ℝ+ is discussed and used to give a representation of the generating functions for discrete completely self-decomposable probabilities on ℤ+. Classes of generalized Γ-convolutions analogous to the multiply self-decomposable probabilities on ℝ+ are studied as well as their discrete counterparts.