When domain decomposition is used in a finite element framework for approximating second order elliptic or parabolic type problems, it becomes appealing to tune the mesh of each subdomain to the local behaviour of the solution. The resulting discretization is then nonconforming, and different approaches have been advocated to match the admissible discrete functions. We introduce here the Mortar Element Method which is an optimal domain decomposition method for the approximation of partial differential equations on non-matching grids and we explain how to compute this method efficiently. We explain how the discretization leads to an algebraic saddle-point problem solved by a preconditioned Finite Element Tearing and Interconnecting (FETI) method. Finally, we provide an extension of the mortar element method to plate problems.