Two hydrogen-rich materials, H₃S and LaH₁₀, 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.