This volume contains the research and expository articles for the courses and talks given at the UIMP-RSME Lluis A. Santalo Summer School, 'Recent Trends in Partial Differential Equations'. The goal of the Summer School was to present some of the many advances that are currently taking place in the interaction between nonlinear partial differential equations and their applications to other scientific disciplines.Oriented to young post-docs and advanced doctoral students, the courses dealt with topics of current interest. Some of the tools presented are quite powerful and sophisticated. These new methods are presented in an expository manner or applied to a particular example to demonstrate the main ideas of the method and to serve as a handy introduction to further study. Young researchers in partial differential equations and colleagues from neighboring fields will find these notes a good addition to their libraries. This is a joint publication of the Real Sociedad Matematica Espanola and the American Mathematical Society.