The search for microscopic models to explain the many superconducting substances has introduced seminal concepts and techniques in many-body physics and in statistical mechanics. The complexity of the high-temperature superconductors has required a remarkable refinement of experimental techniques in order to allow a reliable characterization of the samples, and is partly the reason why so many different microscopic models have so far been proposed. This Enrico Fermi Course on Superconductivity was provided an up-to date presentation of selected experimental and theoretical theories on the (so called) conventional superconductivity and on the high temperature superconductivity. The attention was focused on those reliable measurements which are expected to provide the theory with key constraints, viz: Raman and Infrared Spectroscopy, Nuclear Spin Resonance, Angular Resolved Photoemission Spectroscopy, transport measurements, Josephson effect. The lectures devoted to the overview of the BCS theory and to the discussion of minimal models and of the crossover from BCS to Bose-Einstein condensation may be particularly useful. The remaining part of the program was shared between phonon and non-phonon based mechanisms. On the one hand, special emphasis has been devoted to the breakdown of the Migdal theorem and to polaronic theories. On the other, the book contains an overview of strongly correlated electron theories, including magnetic interactions. A survey of the physics of vortices completes the theoretical part of the lectures.