It is possible to develop analytical solutions for engineering problems based on the equations of equilibrium and compatibility; however, it is necessary to make certain simplifying assumptions regarding material behavior, geometry, and boundary conditions. Numerical procedures such as FD, FE, BE, and energy methods can be developed for the solution of such problems by reducing the simplifying assumptions. It is believed that the FEM possesses certain advantages and generality compared to other methods. Hence, in this chapter, we primarily focus on the FEM for problems involving plane deformations in solids. Limited applications of some other methods (stiffness and energy) are also included.