We report a theoretical study of the physisorption of molecular hydrogen, H2, on a major component of the interstellar dust, namely, the polyaromatic carbonaceous grains. Going beyond the model of the polycyclic aromatic hydrocarbon freeflyers and its theoretical treatment within the super molecule approach, we consider the graphene surface in a Density Functional Theory periodic approach using plane-wave expansions. The physisorption energy of isolated H2 on that flat and rigid support is determined to be attractive by ∼0.75 kcal mol−1 and practically independent of the orientation with respect to the infinite surface. Since this energy is also not affected by the position (over a ring centre, a carbon atom or the middle of a carbon–carbon bond), we can conclude that H2 is able to move freely like a ball rolling on the graphene support. We also investigate the conditions for multiple physisorption. It leads to a monolayer of H2 molecules where the corresponding interaction energy per H2 amounts to a potential depth of ∼1 kcal mol−1, close to the available experimental estimates ranging from 1.1 to 1.2 kcal mol−1. We show that the most energetically favourable coverage, which corresponds to an arrangement of the H2 molecules, the closest possible to the dimer configuration, leads to a surface density of ∼0.8 × 1015 molecule cm−2. Finally, assuming that 15–20 per cent of the interstellar carbon is locked in aromatic systems, one obtains ∼10−5 of the interstellar hydrogen trapped as H2 on such types of surfaces.