We address the problem of reconstructing a real potential $V$ from the Dirichlet-to-Neumann map of a Schrödinger operator $-\Delta + V$ on the boundary of a domain in Euclidean space (the reconstruction aspect of the Calderón problem). This problem is rather involved in general, from both the analytical and numerical points of view as it enjoys poor stability properties. After reviewing the main tools used in proving that potentials are uniquely determined from the D-t-N map, we introduce an object that is obtained in terms of certain matrix elements of the D-t-N map -- the Born approximation -- which is reminiscent of an approximation for the potential that has been extensively studied in the context of inverse scattering theory. We will show a number of interesting analytical properties of the Born approximation in the case of radial potentials in a ball. Among others, how it can be used to factorize the reconstruction problem into a linear, yet ill-conditioned problem (the Hausdorff moment problem) and a nonlinear step which enjoys Hölder stability. If time permits, we will also present a novel algorithm for numerical reconstruction based on this object. This is based on a series of works in collaboration with J.A. Barceló, C. Castro, T. Daudé, C. Meroño, F. Nicoleau, D. Sánchez-Mendoza.