In the talk I will describe some nonlinear severly ill-posed inverse boundary value problems involving elliptic equations and elliptic systems with applications to medical imaging, non destructive testing of materials and seismology. More precisely, one wants to determine some coecient appearing in an elliptic equation or system in a bounded domain \Omega \subset \mathcal R^3 from observations of solutions and its derivatives on \partial \Omega. In particular, I will focus my attention on the conductivity problem, the Gelfand-Calderon problem and the elasticity inverse problem reviewing some of the main results concerning uniqueness and continuous dependence. In the second part of the talk I will concentrate on the issue of continuous dependence, crucial for eective reconstruction, describing some recent results where Lipschitz continuous dependence estimates have been derived in the case of coecients that are nite linear combinations of functions \psi_j ; j = 1; ;N, dened on a Lipschitz partition D_N = \cup_{j-1}^N D_j of \Omega, for example when \psi_j = \chi_{D_j}, j = 1; ;N. This is quite natural having in mind a nite element scheme used in the reconstruction procedure. A crucial role is played by the Lipschitz constant appearing in the estimates and its dependence on the a priori parameters in particular on the mesh size r = r(N) of the partition D_N. I will present some recent results concerning the Gelfand-Calderon problem; in this case an explicit optimal bound of the Lipschitz constant with respect to the mesh size is derived. The results are obtained in collaboration with E. Francini, M. de Hoop, L. Qiu, O. Scherzer and S. Vessella.