abstract: This talk is mainly a report about joint works with Vittoria Pierfelice (Orleans), and also with Maria Vallarino (Milano), about dispersive PDE on hyperbolic spaces (actually Damek-Ricci spaces). Besides we shall give a survey of related results obtained by other authors. The first part of the talk will be devoted to the semilinear Schrödinger equation associated to the Laplace-Beltrami operator and also to a distinguished Laplacian. The second part of the talk will be devoted to the semilinear wave equation. As might be expected, dispersive properties are better in negative curvature and therefore results are stronger than in the Euclidean setting.