The potential of Carbon structures to store hydrogen has been discussed in the literature very controversially. In this contribution we revisit the capabilities for hydrogen storage in graphite based structures and in carbon nanotubes by calculations of interaction potentials and by computer simulations of the free energies and the equilibrium constants for hydrogen adsorption, considering quantum effects. We have shown that a proper manipulation of the interlayer distance between graphene layers can considerbly increase the free energy of molecular hydrogen adsorption. Therefore, proper ways for such manipulation have to be searched and other carbon forms have to be investigated with this respect. Since the discovery of fullerenes and nanotubes several other new forms of carbon have been found (e.g. onions, diamondoids, peapods) or proposed (e.g. metallic carbon, graphyne, scrolls). In addition to pure sp2 and sp3 systems, there have been investigated also mixed sp2 - sp3 connected structures (e.g. isodiamond-graphite hybrids, vacancies of graphite, carbon foams). Here we consider the increase of the interlayer distance in graphite by fullerene intercalation, the so called carbon foams and a fullerite structure concerning their hydrogen storage potential. These nanostructured carbon modifications exhibit a large accessible internal surface and active volume for molecular hydrogen adsorption. I.e., nanostructured carbon modifications have a considerable potential for physisorptive hydrogen storage.