The purpose of the present book is to offer an up-to-date account of the theory of viscosity solutions of first order partial differential equations of Hamilton-Jacobi type and its applications to optimal deterministic control and differential games. The theory of viscosity solutions, initiated in the early 80's by the papers of M.G. Crandall and P.L. Lions CL81, CL83], M.G. Crandall, L.C. Evans and P.L. Lions CEL84] and P.L. Lions' influential monograph L82], provides an - tremely convenient PDE framework for dealing with the lack of smoothness of the value functions arising in dynamic...