We discuss the problems of representation of solutions of applied problems of the theory of elasticity in the form of expansions in some system of base solutions. Particularly, we consider the solutions of two-dimensional problems for orthotropic body and of problems of theory of composite plates and shells. The main difficulties of construction of such solutions may thus be reduced to the problem of determination of the expansion coefficients. That is why, increasing of accuracy of such approximate solutions, improvement of their physical clearness and validity, and estimation of accuracy is of great importance. We propose new methods, which allow us to obtain exact solutions in the form of expansions in canonical domains with application of special systems of functions and biorthogonal expansions. Some aspects of creation of models of media with generalized kinematics are considered. We also discuss some new forms of representation of searched solutions of problems of deformable solids in general three-dimensional statement in the form of expansion into subspaces of kinematic states in energy norms defined by potential energy of deformation.