We present a unified account of a diverse range of problems for which upper and lower bounding principles can be proved. These principles state that a solution of the full problem provides a functional \(J\) with a minimum value among the solutions of one subset of the governing equations; and also provides another functional \(K\) with a maximum value among the solutions of a different subset. The minimum of \(J\) equals the maximum of \(K\). We give explicit illustrations in contexts which include finite linear and nonlinear programming, network theory, optimization and control theory, fluid mechanics, elasticity and plasticity, and other general boundary value problems and operator equations from theoretical physics, economic optimization and elsewhere. We set down in Section 2 a simple characteristic analytical structure which applies to all the problems we consider. These governing conditions may contain equations or inequalities or both, and we describe them as being of either generalized Lagrangian or Hamiltonian form. Our general theory continues in Sections 3 to 8 with an account of the ideas of the Legendre transformation, convex functions and saddle functions and their interrelations, uniqueness theorems and dual extremum principles. The remainder of the paper is concerned with applications (Sections 9 to 15). This general theory is designed to be readable at two distinct levels -- either at the elementary finite-dimensional level of matrix algebra, or at the level of general operators on inner product spaces (for which an Appendix on functional analysis is provided (see p. 187)). The reader may therefore choose the level of his first reading.