We briefly review dispersive transport and treat in more detail the multiple-trapping mechanism. We consider the problem of transient current spectroscopy: Given the experimental current I(t) how does one extract a distribution of rates? Extensive new measurements were made in a well studied polymeric material. Photocurrents in polyvinylcarbarzole were studied over 10 decades in time. An analytical approach is proposed for extracting rate distributions from the measured photocurrents. We find that the trapping-rate distribution is not exponential but "flat" and that it shows a cutoff at low rates. This distribution gives rise to the novel feature of a gradual transition from dispersive to nondispersive charge transport during a single transit. The temperature and field dependence of the cutoff rate rc was studied in the framework of a Poole-Frenkel model.