The field of inverse scattering theory has been a particularly active field in applied mathematics for the past twenty five years.The aim of research in this field has been to not only detect but also to identify unknown objects or "targets" through the use of acoustic,electromagnetic or elastic waves.Although the success of such techniques as ultrasound and x-ray tomography in medical imaging has been truly spectacular,progress has lagged in other areas of application which are forced to rely on different modalities using limited data in complex environments.Until a few years ago,essentially all existing algorithms were based on either a "weak scattering" approximation or on the use of nonlinear optimization techniques.However,as the demands of imaging increased,it became clear that incorrect model assumptions inherent in weak scattering approximations impose severe limitations on when reasonable reconstructions are possible.On the other hand,it was also realized that for many practical applications nonlinear optimization techniques require a priori information that is in general unavailable.Hence in recent years alternative methods for imaging have been developed which avoid incorrect model assumptions but,as opposed to nonlinear optimization techniques,only seek limited information about the scattering object.One such method is the linear sampling method and this talk is designed to be an introductory lecture which presents the main ideas of this new approach to imaging with special emphasis on acoustic and electromagnetic waves. Concerning references,the following surveys are probably the best(I am sending you a reprint of the SIAM Review aricle in the mail today): 1.D. Colton,J.Coyle and P. Monk,Recent developments in inverse acoustic scattering theory,SIAM Review 42(2000),369-414. 2.D. Colton,Inverse acoustic and electromagnetic scattering theory,in Inside Out:Inverse Problems and Applications,G.Uhlmann,editor,MSRI Publications,Vol.47,Cambridge University Press,2003. 3.F. Cakoni and D. Colton,Qualitative Methods in Inverse Scattering Theory,Springer Verlag Series on Interactions of Mechanics and Mathematics,2006.