We present scanning tunneling spectroscopy (STS) investigations of high temperature superconductors illustrating the very unusual characteristics of the superconducting gap, and the intimate relation existing between the superconducting and the normal-state gap (pseudogap). A unique feature of STS is to probe the local density of quasiparticle states bound to individual vortices. The vortex core spectroscopy sheds new light on the pseudogap and the microscopic nature of HTS. The experiments reviewed here suggest that HTS are in a regime very different from BCS, approaching the crossover to Bose-Einstein condensation.