  The classical Bardeen-Cooper-Schrieffer and Eliashberg theories of the
electron-phonon-mediated superconductivity are based on the Migdal theorem,
which is an assumption that the energy of charge carriers, $k{_B}T{_F}$,
significantly exceeds the phononic energy, $\hbar{\omega{_D}} $, of the
crystalline lattice. This assumption, which is also known as adiabatic
approximation, implies that the superconductor exhibits fast charge carriers
and slow phonons. This picture is valid for pure metals and metallic alloys
because these superconductors exhibit $\hbar{\omega{_D}}$/$k{_B}T{_F}<0.01$.
However, n-type doped semiconducting $SrTiO_3$ was the first superconductor
which beyond this adiabatic approximation, because this material exhibits
$\hbar{\omega{_D}}$/$k{_B}T{_F} $~$ 50$. There is growing number of newly
discovered superconductors which also beyond the adiabatic approximation. Here,
leaving apart pure theoretical aspects of nonadiabatic superconductors, we
classified major classes of superconductors (including, elements, A-15 and
Heusler alloys, Laves phases, intermetallics, noncentrosymmetric compounds,
cuprates, pnictides, highly-compressed hydrides and oxygen, and magic-angle
twisted bilayer graphene) by the strength of nonadiabaticity (for which the
ratio of the Debye temperature to the Fermi temperature, $T{_\theta}/T{_F}$, is
used as a criterion for the nonadiabaticity) versus the superconducting
transition temperature, $T{_c}$. The discussion of this classification scheme
and its relation to other known classification counterparts is given.
