An algorithm for deriving IUPAC names of fullerenes is described. Several possible ways for coding of fullerenes are also discussed: the known spiral ring code and its modified version, the Weinberg code, the code derived from the Eulerian trail, the code derived from the Brugesser-Mani theorem, two others based on a Hamiltonian cycle and a Hamiltonian path, and the code based on a path-cotree. A special nomenclature for fullerenes and their derivatives is under consideration in CAS and IUPAC bodies, based upon a spiral passing through all atoms in a molecule. Some isomers relevant for this scheme are pointed out: two highly symmetric isomers with no such spiral and one isomer with such a spiral in only one direction. 1995, American Chemical Society. All rights reserved.