Lévy-type perpetuities being the a.s. limits of particular generalized Ornstein-Uhlenbeck processes are a natural continuous-time generalization of discrete-time perpetuities. These are random variables of the form $S := \int_{(0,\infty)} e^{-X_{s-}} dZ_s$ , where (X, Z) is a two-dimensional Lévy process, and Z is a drift-free Lévy process of bounded variation. We prove an ultimate criterion for the finiteness of power moments of S. This result and the previously known assertion due to Erickson and Maller (2005) concerning the a.s. finiteness of S are then used to derive ultimate necessary and sufficient conditions for the L p-convergence for p > 1 and p = 1, respectively, of Biggins' martingales associated to branching Lévy processes. In particular, we provide final versions of results obtained recently by Bertoin and Mallein (2018).