Physisorption on graphite, graphene, nanotubes, and other graphitic structures has been the subject of numerous studies, partly driven by interest in the nature of order in two-dimensional systems, their phase transitions, and the use of graphitic scaffolds for reversible storage of hydrogen at high volumetric density and low mass. In contrast, physisorption on individual fullerenes or small aggregates of fullerenes has remained largely unexplored, last but not least, because of technical challenges. A summary of recent progress in identifying specific adsorption sites on positively charged C60, C70, and their aggregates is given in this Minireview. Adsorption energies and storage capacities for helium, hydrogen, methane, oxygen, nitrogen, water, and ammonia are determined. Mass spectrometric data reveal the formation of a commensurate phase in which all hollow sites of C60 or C70 are occupied. This phase is identified for all nonpolar molecules, including oxygen, which does not form a commensurate phase on planar graphite. The polar molecules, on the other hand, do not wet fullerenes and they do not form this commensurate phase. A hierarchy of other distinct adsorption sites are identified for nonpolar molecules, namely, groove sites for fullerene dimers and beyond, and dimple sites for fullerene trimers and beyond. Furthermore, evidence is presented for the preferential adsorption of hydrogen and methane in registered sites on fullerene dimers. The interpretation of experimental data that merely count the number of preferred adsorption sites is aided by molecular dynamics simulations, which utilize interaction potentials derived from ab initio calculations to determine adsorption energies.