The basic field equations in theory of elasticity have been developed in the previous chapters. These results comprise a system of differential and algebraic relations among the stresses, strains and displacements that express particular physics at all points within the body under investigation. In this chapter we now would like to complete the general formulation by first developing boundary conditions appropriate for use with the field equations. These conditions specify the physics that occur on the boundary of the body, and generally provide the loading inputs that physically create the interior stress, strain and displacement fields. Although the field equations are the same for all problems, boundary conditions are different for each problem. Therefore, proper development of boundary conditions is essential for problem solution, and thus it is important to acquire a good understanding of such development procedures. Combining field equations with boundary conditions then establishes the fundamental boundary value problems of the theory. This eventually leads us into two different formulations: one in terms of displacements and the other in terms of stresses. Because boundary value problems are difficult to solve, different strategies have been developed to solve the problems. In this chapter, we review in a general way several of these strategies.