The diffraction of elastic waves by inclusions, cavities and cracks has been the subject of numerous studies in recent years. Because of analytical difficulties, however, most of these studies have dealt with spherical or circular inclusions and straight or circular cracks. In a series of recent papers we have shown that for long wavelengths it is possible to obtain an asymptotic expansion of the scattered field by the method of matched asymptotic expansions. These papers have dealt with ellipsoidal and elliptic inclusions. We also have been able to use this technique to solve scattering problems in a half-space. In this paper we present briefly the results that we have obtained in the course of this investigation. The results include scattering by ellipsoidal inclusions in three dimensions and elliptic cylinders in two dimensions, by buried circular cavities and elliptical inclusions in a halfspace, and by an edge crack. In the context of the last problem it is shown that MAE together with analytic function techniques can be used to solve many (not necessarily straight) crack problems in two dimensions.