Hydrogen-rich compounds are considered most likely to achieve room-temperature superconductivity since the critical temperature (T c) above 250 K was observed in lanthanum hydride. Exploring the high-temperature superconductivity in rare-earth metal hydrides becomes very interesting. Based on the particle swarm optimization for crystal structures and first-principles calculations, we investigate the crystal structures, phase stability, metallization, and possible superconducting properties of terbium hydride (TbH n , n = 1 – 12) under pressure. Our results show that terbium hydride is a potential high-temperature superconductor under high pressures. It stably exists at different pressure conditions by adjusting the H content. Specifically, the H atomic cage structure can be observed in most terbium hydrides, and the number of H atoms in the cage sublattice increases with the stoichiometry of H in TbH n . We demonstrate that the high T c value is closely related to this cage sublattice and it increases with increasing H content in terbium hydride. The highest T c above 270 K is predicted in TbH10 at 250 GPa for Fm3̅m and 310 GPa for R3̅m space group. This result indicates that the superconductivity with T c close to or beyond lanthanum hydride can be achieved in other rare-earth metal hydrides.