The evolution of superfluidity in a quantum gas of interacting fermionic particles crossing over from a Bose-Einstein Condensation (BEC) of composite bosons to a Bardeen-Cooper-Schrieffer (BCS) superfluidity of Cooper pairs, is still an open problem reserving surprises. In the first part of the talk, I will discuss the original idea from the seminal works by Leggett and by Nozières and Schmitt-Rink, how it has provided a crucial many-body paradigm to understand the physics of more complex systems, such as high-temperature superconductors, and how quantum gases represent a formidable, combined experimental and theoretical platform to probe the crossover idea against different microscopic models under highly controllable conditions.
In the second part,I will explore different realizations of model hamiltonians and quantitatively discuss the extent to which universal behavior manifests in the relevant observables like chemical potential, condensate fraction, or critical temperature, independently of the details of the microscopic interaction. I will consider fermions interacting via (i) a short-range well-barrier potential,(ii) a potential with tunable strength and range, in the presence of spin-orbit coupling originated by an artificial magnetic field, below and above the threshold for topological changes of the Fermi sphere, and (iii) a Fano-Feshbach resonance mechanism with variable resonance width between extreme narrow and broad limits. In this latter context, I will briefly present a novel unifying self-consistent theory, including the fluctuations beyond mean field in non-perturbative manner.