  Two hydrogen-rich materials, H$_3$S and LaH$_{10}$, synthesized at megabar
pressures, have revolutionized the field of condensed matter physics providing
the first glimpse to the solution of the hundred-year-old problem of room
temperature superconductivity. The mechanism underlying superconductivity in
these exceptional compounds is the conventional electron-phonon coupling. Here
we describe recent advances in experimental techniques, superconductivity
theory and first-principles computational methods which have made possible
these discoveries. This work aims to provide an up-to-date compendium of the
available results on superconducting hydrides and explain how the synergy of
different methodologies led to extraordinary discoveries in the field. Besides,
in an attempt to evidence empirical rules governing superconductivity in binary
hydrides under pressure, we discuss general trends in the electronic structure
and chemical bonding. The last part of the Review introduces possible
strategies to optimize pressure and transition temperatures in conventional
superconducting materials as well as future directions in theoretical,
computational and experimental research.
