In this paper, we consider some geometric inverse problems for the Lame system. A first motivation comes from fromelastography,a technique that allows to identify the elastic properties of tissues and can serve, among other things, to detect a tumor from non-invasive (external) measurements. A second one is to identify a rigid structure in an elastic medium. We prove a uniqueness result, we introduce some iterative methods for the reconstruction problem and we present the results of several numerical experiments.