Resolving the challenge of hydrogen storage is considered the last frontier towards the transition to a worldwide energy network in which hydrogen can be used as an efficient and carbon-free energy carrier. The technical difficulties of storing hydrogen efficiently in a compressed gas or cryogenic liquid form have directed the global scientific community on investigating solid materials with the ability to physically or chemically bind hydrogen and then reversibly release it by varying the operating temperature and pressure. Physical adsorption is one of the most attractive methods of storing hydrogen in porous materials with large specific areas and pore volumes as well as nanometer-sized pore widths, as it reduces significantly the large volume occupied by gaseous hydrogen, is completely reversible and allows fast adsorption/desorption kinetics. In this thesis, a large variety of carbon-based and hybrid materials, including carbon nanotubes, graphene oxide sponges and foams, few-layer graphene flakes, activated carbon cloths and metal-organic frameworks, were systematically studied for their hydrogen adsorption performance under compression both at cryogenic and ambient temperatures. Porosity-related structural features, such as the pore size distribution and the average pore size, seem to critically influence the hydrogen adsorption behavior of these materials.