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In many problems of applied mathematics, science, engineering or economics, an energy expenditure or its analogue can be approximated by upper and lower bounds. This book provides a unified account of the theory required to establish such bounds, by expressing the governing conditions of the problem, and the bounds, in terms of a saddle functional and its gradients. There are several features, including a chapter on the Legendre dual transformation and some of its singularities. Many substantial examples and exercises are included, especially from the mechanics of fluids, elastic and plastic solids and from optimisation theory. The saddle functional viewpoint gives the book a wide scope. The treatment is straightforward, the only prerequisite being a basic knowledge of the calculus of variations. Part of the book is based on final-year undergraduate courses. This is developed into an account which will interest a wide range of students and professionals in applied mathematics, engineering, physics and operations research.