Summary. Predicting future rates of species discovery and the number of species remaining are important in efforts to preserve biodiversity, discussions on the rate of species extinction and comparisons on the state of knowledge of animals and plants of different taxa. Data on discovery dates of species in 32 European marine taxa are analysed by using a class of thinned temporal renewal process models. These models allow for both underdispersion and overdispersion with respect to the non-homogeneous Poisson process. An approach for implementing Bayesian inference for these models is described that uses Markov chain Monte Carlo simulation and that is applicable to other types of thinned process. Predictions are made on the number of species remaining to be discovered in each taxon.