Palladium hydride was discovered more than 150 years ago and remains one of the most-studied interstitial metal hydrides because of the richness of its physical behaviours, which include ordered phases and anomalous properties at temperatures below 100 K, a superabundant-vacancy (SAV) phase with stoichiometry Pd3H4 formed at high temperature and pressure, and quenching of the enhanced Pauli paramagnetism of palladium. One of the most fascinating properties of palladium hydride is superconductivity at about 10 K without external pressure, in contrast to the newly-discovered polyhydride room-temperature superconductors that require megabar pressures. Moreover, the superconductivity exhibits an inverse isotope effect. Remarkably, modern first-principles approaches are unable to accurately predict the superconducting transition temperature by calculating the electron–phonon coupling constant within Migdal-Eliashberg theory. Anharmonicity of the hydrogen site potential is a key factor and poses a great challenge, since most theoretical approaches are based on the harmonic approximation. This review focuses on the electron and phonon band structures that underpin all such calculations, with palladium as a reference point. While the electron band structures of palladium and its monohydride are uncontroversial, the phonon band structure of palladium hydride in particular is problematic, with a realistic treatment of anharmonicity required – and largely yet to be achieved – to reproduce the results of inelastic neutron scattering experiments. In addition to the monohydride and SAV phases, possible higher hydrides are surveyed and the origin of the famous “50-K” anomaly in specific heat and other physical properties is critically reviewed.