The possibility of π-electron ring currents in C60 has been of interest since the initial identification of the fullerenes and the recent synthesis of these compounds has provided an experimental impetus to magnetic studies. We calculated a vanishingly small π-electron ring-current magnetic susceptibility for C60 and this prediction has recently received experimental verification. We attributed this behavior to a near cancellation of the diamagnetic and Van Vleck paramagnetic terms. The higher fullerenes may become available for study in the near future and recent work by Diedrich et al. (Science, 252 (1991) 548) has led to the isolation of C84. Although the structure of this molecule has not yet been identified, Raghavachari and Rohlfing (J. Phys. Chem., 95 (1991) 3457) have reported ab initio calculations of the structures of C60, C70 and two isomers of C84. We therefore decided to study these structures with the three-dimensional finite-field London theory previously employed to study C60 and C70 with idealized geometries. The π-electron ring-current magnetic susceptibilities of the fullerenes included in the present study do not monotonically increase toward the graphite value. Both isomers of C84 show small average diamagnetic susceptibilities. There is a large anisotropy associated with the π-electron diamagnetic susceptibility of C84 (D6h). With the exception of C70, all of the molecules show a strongly enhanced diamagnetism in the 6-state and a weak diamagnetism in the 12-state. © 1993.