Book excerpt: This dissertation presents a new hybrid elasticity and finite element method for general non-conforming contact problems. It is an iterative numerical procedure, which has distinct advantage over classical theory of contact since general geometrical and loading profiles with friction can be treated. And over the traditional contact solution approach in finite element method, it eliminates the use of gap elements and therefore the non-linearity in the solution enhancing accuracy and efficiency of the solution. For the two dimensional problems the equations were derived for a triangular pressure element. While for three-dimensional problems, the analytical solution was derived from Boussinesq and Cerruti equations for conical pressure elements, from which a semi-analytical approach to three-dimensional contact problems with friction was evolved to find the extent of the contact area and the loading distribution over this area together with the extent of the stick and slip zones. The linear static finite element method was then used to find the displacements and stresses throughout the two bodies in contact, which were analyzed separately with the knowledge of the area of contact and the traction over such an area. The method was used for non-conforming bodies but it can be extended to include different kinds of contact problems. The basic theory was presented for both two and three-dimensional non-conforming contact problem, together with algorithm, and numerical examples that showed the accuracy, efficiency and robustness of the method.