Uniqueness in Inverse Obstacle Scattering

The inverse problem we consider in this survey is to determine the shape of an obstacle from the knowledge of the far field pattern for the scattering of time-harmonic waves. We will concentrate on uniqueness issues, i.e., we will investigate under what conditions an obstacle and its boundary condition can be identified from a knowledge of its far field patterns for incident plane waves. We will review some classical and some recent results and draw attention to open problems.