This book provides a modern and comprehensive presentation of a wide variety of problems arising in nonlinear analysis, game theory, engineering, mathematical physics and contact mechanics. It includes recent achievements and puts them into the context of the existing literature. The volume is organized in four parts. Part I contains fundamental mathematical results concerning convex and locally Lipschits functions. Together with the Appendices, this foundational part establishes the self-contained character of the text. As the title suggests, in the following sections, both variational and topological methods are developed based on critical and fixed point results for nonsmooth functions. The authors employ these methods to handle the