Contains the Lecture Notes of the school on this subject held in S. Momme' (Pistoia) in May 2002. The aim of the school was to introduce a wide audience of mathematicians, physicists and engineers to some advanced topics in nonlinear analysis and applications to physiscal sciences. This work covers some aspects of nonlinear analysis and applications such as fractal analysis, variational methods in nonlinear field equations, self contact problems in lasticity, qualitative properties of solutions of variational problems.