Carbon materials have proven to be a suitable choice for hydrogen storage and, recently, for hydrogen compression. Their developed textural properties, such as large surface area and high microporosity, are essential features for hydrogen adsorption. In this work, we first review recent advances in the physisorption characterization of nanoporous carbon materials. Among them, approaches based on the density functional theory are considered now standard methods for obtaining a reliable assessment of the pore size distribution (PSD) over the whole range from narrow micropores to mesopores. Both a high surface area and ultramicropores (pore width < 0.7 nm) are needed to achieve significant hydrogen adsorption at pressures below 1 MPa and 77 K. However, due to the wide PSD typical of activated carbons, it follows from an extensive literature review that pressures above 3 MP are needed to reach maximum excess uptakes in the range of ca. 7 wt.%. Finally, we present the adsorption–desorption compression technology, allowing hydrogen to be compressed at 70 MPa by cooling/heating cycles between 77 and 298 K, and being an alternative to mechanical compressors. The cyclic, thermally driven hydrogen compression might open a new scenario within the vast field of hydrogen applications.