High-Temperature superconductivity was observed first for the cuprates (1986) and later also in Fullerene’s compounds (1991). A particular characteristic of new superconductive Fullerene compounds (C60) is the small value of Fermi energy (EF=0.2 eV), that is of the same order of phononic energies. This situation leads to the breakdown of Migdal’s theorem (Born-Oppenheimer approximation) that is the basis of the standard theory of superconductivity. In particular, it is striking the comparison of the C60 compounds (Tc = 35 K for Rb3C60 ) with the analogous graphite compounds for which Tc = 0.2 K. Many properties of the two systems are similar, but the graphite compounds satisfy Migdal’s theorem while the C60 compounds do not. The situation for the oxides appears less clear because the nature of the mediator is not yet been established. While for C60 compounds the large isotope effect clearly points to a phononic mechanism, the isotope effect for the oxides ranges from large values in the case of non optimized materials to very small values for the materials with the highest values of Tc. Nevertheless, also for the oxides the value of the effective Fermi energy is rather small and, independently on the nature of the bosonic mediator, one cannot expect Migdal’s theorem to hold. A similar situation can be found also in the organic superconductors.

According to certain empirical classifications, like the one proposed by Uemura and collaborators, these three classes of materials: oxides, fullerene compounds and organics seem to belong to the same broad family of anomalous superconductors, well separated from the family of the standard BCS materials like Al and Pb. An important common element of all these three classes is the fact that the carrier density is extremely small, ten or twenty times smaller than in usual metals. This property appears quite negative from the perspective of usual BCS superconductivity because it leads to a small density of states and it reduces the screening of the Coulomb repulsion. Therefore, even for the systems like the C60 compounds and the non optimized oxides in which the pairing is certainly via phonons, one has to look for a new mechanism in which the small carrier density can play a positive role. In this respect, the nonadiabatic effects arising from the breakdown of Migdal’s theorem appear to be common to the three classes and allow us to interpret in positive the small carrier density as a necessary element related to the small value of the Fermi energy.

It is therefore necessary to generalize the theory of superconductivity by including vertex corrections and other non-adiabatic effects. This situation is more complex than usual one, and can give, under some conditions, high critical temperatures and several other peculiar effects. The question is now the identification of physical conditions which can lead to favourable situations. In particular, it seems to be relevant the role of electronic correlation. Finally it has to be reminded that Fermi energy is small also for oxides materials, and therefore these considerations can have a more general character. Of course, there are many other important effects, expecially for the oxides. The strong electronic correlation is certainly necessary to understand the phase diagram and also for the C60 compounds the behavior as a function of doping is not understandable from the point of view of band filling. This situation, hovewer, is not surprising because, if the Fermi energy is small, the stability of the system in the Fermi liquid phase is precarious and small changes of parameters can lead to instabilities towards magnetism, Mott-Hubbard insulators, Jahn-Teller distortions, charge density waves, stripes etc. Therefore, correlation is essential to understand the phase diagram of the oxides and their magnetic properties. However, once the energy balance of the interaction favours, in a certain range of parameters, a metallic phase then it is not obvious at all that correlation effects are also at the essence of the pairing mechanism of superconductivity.

From this point of view, we study the properties of a strongly correlated metal that does not satisfy Migdal’s theorem. In general, this is quite a difficult task, so we first focusing on the breakdown of Migdal’s theorem and then add the effect of correlations . For the mediators we nominally consider phonons but most of the results can be easily extended to other mediators. However, even with only phonons, the generalized theory is already so reach and complex that it is capable to provide a natural interpretation to several anomalies of both the superconducting and the normal state. At the moment we are looking for specific effects for which one could formulate a theoretical prediction to be tested experimentally. Effects of this type are anomalous isotope effects in the normal and SC phases, the role of nonmagnetic impurities on Tc and the Pauli susceptibility. At least for C60 we can also formulate a schematic material engineering to optimize Tc and we are contacting experimental groups to test these ideas.