The results of an accurate density-functional study of the structure, energetics and electronic structure of Pt-n clusters (with n = 13, 38, and 55) are presented. For Pt-38, a truncated octahedral geometry is considered; for Pt-13 and Pt-55, icosahedral, truncated decahedral, and cuboctahedral geometries are considered. In each case, the structure of the neutral and positively and negatively charged clusters is fully optimized within the given symmetry group. For Pt-13, allowing symmetry breaking starting from the symmetrical structures derives additional local minima. The computational procedure is thoroughly tested to keep numerical accuracy under control. From the electronic structure point of view, it is found that these systems start developing metallic characteristics, with ionization introducing small changes. From the structural point of view, for Pt13 the icosahedral configuration is not favored, whereas it becomes the ground state for Pt55, in agreement with the predictions of atom-atom potentials. Moreover, the lowest energy configuration of Pt13 is a symmetry-broken D-4h one, while for Pt-38 and Pt-55 a peculiar rearrangement is found, corresponding to an expansion (reconstruction) of the atoms lying on (111) or (100) faces.