Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this text provides a comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics. Topics covered include the classical (minimax) mono-duality of convex static equilibria, the beautiful bi-duality in dynamical systems, the tri-duality in non-convex problems and the complicated multi-duality in general canonical systems. A sequential canonical dual transformation method for solving nonlinear problems is developed heuristically and illustrated by use of examples as well as extensive applications of nonlinear systems. This includes differential equations, variational problems and inequalities, constrained global optimization, multi-well phase transitions, non-smooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and non-convex dynamical systems. With coherent exposition, the work fills a large gap between the mathematical and engineering sciences. It shows how to use formal language and duality methods to model natural phenomena, to construct intrinsic frameworks in different fields and to provide ideas, concepts and powerful methods for solving non-convex, non-smooth problems arising naturally in engineering and science. An appendix provides some necessary background from elementary functional analysis. The book should be a useful resource for students and researchers in applied mathematics, physics, mechanics and engineering. The whole volume or selected chapters can also be recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the ideas of optimization and variational methods are employed.