The computational and experimental exploration of the phase diagrams of binary hydrides under high pressure has uncovered phases with novel stoichiometries and structures, some which are superconducting at quite high temperatures. Herein we review the plethora of studies that have been undertaken in the last decade on the main group and transition metal hydrides, as well as a few of the rare earth hydrides at pressures attainable in diamond anvil cells. The aggregate of data shows that the propensity for superconductivity is dependent upon the species used to "dope" hydrogen, with some of the highest values obtained for elements that belong to the alkaline and rare earth, or the pnictogen and chalcogen families.