The talk provides an overview of some recent work on random probability measure vectors and their role in Bayesian statistics. Indeed, dependent nonparametric priors are useful tools for drawing inferences on data that arise from different, though related, studies or experiments and for which the usual exchangeability assumption is not satisfied. The presentation will focus on models based on completely random measures, or suitable transformations thereof, that have an additive, a hierarchical and a nested structure. Some of their distributional properties relevant for prediction will be discussed. These theoretical results are, then, used for devising Markov chain Monte Carlo algorithms that will be implemented within some illustrative examples for analyzing data in the contexts of species sampling problems and survival analysis.