The emergence of a Hydrogen Economy will require the development of new media capable of safely storing hydrogen in a compact and light weight package. Metal hydrides and complex hydrides, where hydrogen is chemically bonded to the metal atoms in the bulk, offer some hope of overcoming the challenges associated with hydrogen storage. The objective is to find a material with a high volumetric and gravimetric hydrogen density that can also meet the unique demands of a low temperature automotive fuel cell. Currently, there is considerable effort to develop new materials with tunable thermodynamic and kinetic properties. This tutorial review provides an overview of the different types of metal hydrides and complex hydrides being investigated for on-board (reversible) and off-board (non-reversible) hydrogen storage along with a few new approaches to improving the hydrogenation-dehydrogenation properties.