The inverse problem we consider is to determine the shape of an obstac
le from a knowledge of the far field pattern for the scattering of inc
ident time-harmonic acoustic (or electromagnetic) plane waves. It occu
rs in a variety of applications and is difficult to solve since it is
nonlinear and improperly posed. In this survey we shall describe two i
ntegral equation methods for the approximate numerical solution of the
inverse obstacle scattering problem in the resonance region.