The Dirichlet process represents the cornerstone of Bayesian Nonparametrics and is used as main ingredient in a wide variety of models. The many generalizations of the Dirichlet proposed in the literature aim at overcoming some of its limitations and at increasing the models' flexibility. In this talk we provide an overview of a large set of such generalizations by using completely random measures as a unifying concept. All the considered models can be seen as suitable transformations of completely random measures and this allows to highlight interesting distributional structures they share a posteriori in several statistical problems ranging from density estimation and clustering to survival analysis and species sampling. Furthermore, we discuss some natural approaches, which rely on additive, hierarchical and nested structures, to derive dependent versions of Bayesian nonparametric models derived from completely random measures.