An initial observation of the formation of WH under pressure from W gaskets surrounding hydrogen in diamond anvil cells led to a theoretical study of tungsten hydride phases. At P = 1 atm no stoichiometry is found to be stable with respect to separation into the elements, but as the pressure is raised WH(n) (n = 1-6, 8) stoichiometries are metastable or stable. WH and WH(4) are calculated to be stable at P > 15 GPa, WH(2) becomes stable at P > 100 GPa and WH(6) at P > 150 GPa. In agreement with experiment, the structure computed for WH is anti-NiAs. WH(2) shares with WH a hexagonal arrangement of tungsten atoms, with hydrogen atoms occupying octahedral and tetrahedral holes. For WH(4) the W atoms are in a distorted fcc arrangement. As the number of hydrogens rises, the coordination of W by H increases correspondingly, leading to a twelve-coordinated W in WH(6). In WH(8) H(2) units also develop. All of the hydrides considered should be metallic at high pressure, though the Fermi levels of WH(4) and WH(6) lie in a deep pseudogap. Prodded by these theoretical studies, experiments were then undertaken to seek phases other than WH, exploring a variety of experimental conditions that would favor further reaction. Though a better preparation and characterization of WH resulted, no higher hydrides have as yet been found.