Physisorption of hydrogen in nanoporous materials offers an efficient and competitive alternative for hydrogen storage. At low temperatures (e.g. 77 K) and moderate pressures (below 100 bar) molecular H2 adsorbs reversibly, with very fast kinetics, at high density on the inner surfaces of materials such as zeolites, activated carbons and metal-organic frameworks (MOFs). This review, by experts of Task 40 ‘Energy Storage and Conversion based on Hydrogen’ of the Hydrogen Technology Collaboration Programme of the International Energy Agency, covers the fundamentals of H2 adsorption in nanoporous materials and assessment of their storage performance. The discussion includes recent work on H2 adsorption at both low temperature and high pressure, new findings on the assessment of the hydrogen storage performance of materials, the correlation of volumetric and gravimetric H2 storage capacities, usable capacity, and optimum operating temperature. The application of neutron scattering as an ideal tool for characterising H2 adsorption is summarised and state-of-the-art computational methods, such as machine learning, are considered for the discovery of new MOFs for H2 storage applications, as well as the modelling of flexible porous networks for optimised H2 delivery. The discussion focuses moreover on additional important issues, such as sustainable materials synthesis and improved reproducibility of experimental H2 adsorption isotherm data by interlaboratory exercises and reference materials.