  Superconductivity is traditionally viewed as a low-temperature phenomenon.
Within the BCS theory this is understood to result from the fact that the
pairing of electrons takes place only close to the usually two-dimensional
Fermi surface residing at a finite chemical potential. Because of this, the
critical temperature is exponentially suppressed compared to the microscopic
energy scales. On the other hand, pairing electrons around a dispersionless
(flat) energy band leads to very strong superconductivity, with a mean-field
critical temperature linearly proportional to the microscopic coupling
constant. The prize to be paid is that flat bands can generally be generated
only on surfaces and interfaces, where high-temperature superconductivity would
show up. The flat-band character and the low dimensionality also mean that
despite the high critical temperature such a superconducting state would be
subject to strong fluctuations. Here we discuss the topological and
non-topological flat bands discussed in different systems, and show that
graphite is a good candidate for showing high-temperature flat-band interface
superconductivity.
