In the last ten years several numerical methods have been developed for the solution of elastic wave scattering problems that have found application in quantitative flaw definition. Before the development of these methods, due to the complexity of Navier’s equation which governs wave motion in an elastic continuum, numerical results were available only for circular cylinders and spheres. The elastic wave equation is separable only in polar and spherical coordinates. For other geometries, three types of numerical methods have been developed. They were all originally developed for acoustic and electromagnetic problems governed by the scalar and vector wave equations respectively.