The superconductivity in iron-based, magnesium diborides, and other novel
high-Tc superconductors, possibly including the recently discovered
superconducting hydrogen disulfide, has a strong multi-band and multi-gap
character and recent experiments support the possibility for a BCS-BEC
crossover induced by strong-coupling and proximity of the chemical potential
to the band edge of one of the bands, with evidences for Lifshitz transitions
associated with changes in the Fermi surface topology . Here we study the
simplest model which accounts for the BCS-BEC crossover in a
two-band / two-gap superconductor, considering tunable interactions.
When the gap is of the order of the local chemical potential, superconductivity
is in the crossover regime of the BCS-BEC crossover and the Fermi surface of
the small band is smeared by the gap opening. In this situation,
small and large Cooper pairs coexist in the condensate, which is the
optimal condition for very high-Tc superconductivity.
Using available experimental data, our analysis shows that iron-based
superconductors have the partial condensate of the small Fermi surface which
is in the BCS-BEC crossover regime , supporting in this way
the ARPES findings. We also discuss different systems in which
the multigap and multiband BCS-BEC crossover can be realized, pointing toward
very high-Tc superconductivity. Here we consider in detail the superconducting stripes in which shape resonances and multigap at
the band edge play a cooperative role in enhancing superconductivity in the
crossover regime . We focus on a
key prediction of the above discussed physics: the isotope effect of the
superconducting critical temperature in the vicinity of a Lifshitz transition,
which has a unique dependence on the energy distance between the chemical
potential and the Lifshitz transition point. Comparisons with available
experimental data for superconducting cuprates and hydrogen disulfide will be
discussed . The recently discovered high-Tc superconductivity in
potassium-doped para-terphenyl will be finally reviewed .
 D. Innocenti et al, Phys. Rev. B 82, 184528 (2010).
International Network MultiSuper: http://www.multisuper.org
 A. Guidini and A. Perali, Supercond. Sci. Technol. 27, 124002 (2014).
 A. Perali, A. Bianconi, A. Lanzara, N.L. Saini,
Solid State Comm. 100, 181, (1996).
 A. Perali, D. Innocenti, A. Valletta, A. Bianconi,
Supercond. Sci. Technol. 25, 124002 (2012).
 M. V. Mazziotti, A. Valletta, G. Campi, D. Innocenti,
A. Perali, A. Bianconi,
arXiv:1705.09690, accepted in Europhy. Lett.