In the first two sections discussion is restricted to the contact and crack problems associated with circular-shaped discs. In particular, the effect of dislocation in front of a circular punch and crack is considered in this chapter. The reader is introduced to the new alternate approach of Fabrikant to solve circular crack and punch problems. The rest of the chapter is devoted to detailed analysis to elliptic-shaped planar surface. Unlike usual approach solution has been done in Cartesian co-ordinate system so that the results can be easily adapted to the layered media. The reader is introduced to the basic formulation of the Hertzian contact theory as well as the elliptic crack problem in terms of the coupled system of dual integral equation. By a suitable transformation in the cylindrical coordinate system, the coupled system of equation reduces to an infinite system of equations which are solved for polynomial loading using the analysis made in Chapter 2. Unlike previous workers, the complete displacement and stress field are obtained not only on the crack surface but also at all interior points. Possible applications to real life problems are discussed.