The book covers three main topics: the classical theory of linear elasticity, the mathematical theory of composite elastic structures, as an application of the theory of elliptic equations on composite manifolds developed by the first author, and the finite element method for solving elastic structural problems. The authors treat these topics within the framework of a unified theory. The book carries on a theoretical discussion on the mathematical basis of the principle of minimum potential theory. The emphasis is on the accuracy and completeness of the mathematical formulation of elastic structural problems. The book will be useful to applied mathematicians, engineers and graduate students. It may also serve as a course in elasticity for undergraduate students in applied sciences.