This book has been awarded the Ferran Sunyer i Balaguer 2005 prize. The aim of this monograph is to discuss several elliptic problems on Rn with two main features: they are variational and perturbative in nature, and standard tools of nonlinear analysis based on compactness arguments cannot be used in general. For these problems, a more specific approach that takes advantage of such a perturbative setting seems to be the most appropriate. The first part of the book is devoted to these abstract tools, which provide a unified frame for several applications, often considered different in nature.Such applications are discussed in the second part, and include semilinear ellipticproblems on Rn, bifurcation from the essential spectrum, the prescribed scalarcurvature problem, nonlinear Schrödinger equations, and singularly perturbedelliptic problems in domains. These topics are presented in a systematic andunified way.