The development of a clean hydrogen economy will aid a smooth transition from fossil fuels which is required to stem the environmental impact and economic instability caused by oil dependency. For vehicular application, in addition to being cheap and safe, a commercial hydrogen store must contain a certain weight percentage of hydrogen to provide a reasonable range (~300 miles). It must also be able to release hydrogen under near-ambient conditions (80-120°C) and have a reasonable cycling capacity (~1000 cycles). The primary motivation of this thesis is to gain a fundamental understanding into the sorption processes of hydrogen on carbon- and aluminium-based materials to improve their hydrogen storage capacity. The sorption processes of hydrogen on mechanically milled graphite have been investigated, primarily using Electron Spin Resonance Spectroscopy and Inelastic Neutron Scattering. An investigation into the storage properties of tetrahydroaluminates, primarily NaAlH₄ and LiAlH₄, is performed in the presence and absence of a catalyst, and a new phase of NaAlH₄ is observed prior to its decomposition. Variable temperature neutron and synchrotron diffraction, in conjunction with gravimetric and mass spectroscopy data were obtained for several mixtures of tetrahydroaluminates and alkali amides and the hydrogen desorption processes are shown to be quite different from the constituent materials. The structure of Ca(AlH₄)₂ has been experimentally determined for the first time and a complete set of equations describing its decomposition pathway is given.