We determine the saturation storage density for hydrogen on several metal-organic framework (MOF) and covalent-organic framework (COF) materials by spillover. We use density functional theory on periodic frameworks to achieve reliable and accurate predictions for these materials. We find that one hydrogen can be stored at each C atom of the linker, and an additional H for each CO 2 group. For IRMOF-1 and IRMOF-8, we find reasonable agreement with the experimental results. For other materials, such as COF-1 and MOF-177, we find that the experiments could be dramatically improved. We also predict the gravimetric and volumetric storage densities for several new materials, including IRMOF-9, IRMOF-993, MIL-101, PCN-14, COF-1, and COF-5, which appear very promising. We find gravimetric storage densities up to 5.5 wt % and volumetric storage densities up to 44 g/L.