  The BCS-BEC crossover realized experimentally with ultra-cold Fermi gases may
be considered as one of the important scientific achievements occurred during
the last several years. The flexibility for operating on these systems on the
experimental side and the full control of the relevant system degrees of
freedom on the theoretical side make quite stringent at a fundamental level the
comparison between the experimental data and the corresponding theoretical
calculations. Here, we briefly survey recent theoretical advances resting on a
diagrammatic approach at equilibrium that improves in a systematic way on the
widely used t-matrix approach, yielding a quite good comparison between theory
and experiments for several physical quantities of interest. It is proposed
that the physical phenomena underlying this theoretical approach may also be
relevant to the superconducting phase of condensed-matter materials which
cannot be described by the standard BCS theory.
