We report the H2 uptake properties of six covalent organic frameworks (COFs) from first-principles-based grand canonical Monte-Carlo simulations. The predicted H2 adsorption isotherm is in excellent agreement with the only available experimental result (3.3 vs 3.4 wt % at 50 bar and 77 K for COF-5), also reported here, validating the predictions. We predict that COF-105 and COF-108 lead to a reversible excess H2 uptake of 10.0 wt % at 77 K, making them the best known storage materials for molecular hydrogen at 77 K. We predict that the total H2 uptake for COF-108 is 18.9 wt % at 77 K. COF-102 shows the best volumetric performance, storing 40.4 g/L of H2 at 77 K. These results indicate that the COF systems are most promising candidates for practical hydrogen storage.