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The basic facts and fundamental properties of binary fullerene cocrystals are reviewed, focusing especially on solvates and salts of Buckminsterfullerene (C60), and hydrates of hydrophilic C60 derivatives. The examined properties include the lattice structure and the presence of orientational disorder and/or rotational dynamics (of both fullerenes and cocrystallizing moieties), thermodynamic properties such as decomposition enthalpies, and charge transport properties. Both thermodynamic properties and molecular orientational disorder shed light on the extent of intermolecular interactions in these binary solid-state systems. Comparison is carried out also with pristine fullerite and with the solid phases of functionalized C60. Interesting experimental findings on binary fullerene cocrystals include the simultaneous occurrence of rotations of both constituent molecular species, crystal morphologies reminiscent of quasicrystalline behavior, the observation of proton conduction in hydrate solids of hydrophilic fullerene derivatives, and the production of super-hard carbon materials by application of high pressures on solvated fullerene crystals.