Band theory and the BCS-Eliashberg theory of superconductivity are arguably the most successful theories of condensed matter physics by the breadth and subtlety of the phenomena they explain. Experimental discoveries, however, clearly signal their failure in certain cases. Around 1940, it was discovered that some materials with an odd number of electrons per unit cell, for example NiO, were insulators instead of metals, a failure of band theory [1]. Peierls and Mott quickly realized that strong effective repulsion between electrons could explain this (Mott) insulating behaviour [2]. In 1979 and 1980, heavy fermion [3] and organic [4] superconductors were discovered, an apparent failure of BCS theory because the proximity of the superconducting phases to antiferromagnetism suggested the presence of strong electron-electron repulsion, contrary to the expected phonon-mediated attraction that gives rise to superconductivity in BCS. Superconductivity in the cuprates [5], in layered organic superconductors [6, 7], and in the pnictides [8] eventually followed the pattern: superconductivity appeared at the frontier of antiferromagnetism and, in the case of the layered organics, at the frontier of the Mott transition [9, 10], providing even more examples of superconductors falling outside the BCS paradigm [11, 12]. The materials that fall outside the range of applicability of band and of BCS theory are often called strongly correlated or quantum materials. They often exhibit spectacular properties, such as colossal magnetoresistance, giant thermopower, high-temperature superconductivity etc.

The failures of band theory and of the BCS-Eliashberg theory of superconductivity are in fact intimately related. In these lecture notes, we will be particularly concerned with the failure of BCS theory, and with the understanding of materials belonging to this category that we call strongly correlated superconductors. These superconductors have a normal state that is not a simple Fermi liquid and they exhibit surprising superconducting properties. For example, in the case of layered organic superconductors, they become better superconductors as the Mott transition to the insulating phase is approached [13].