The π-electron ring current magnetic susceptibilities and endohedral chemical shifts of the fullerenes are calculated with the London theory. The diamagnetism calculated for the fullerenes that have been characterized to date does not show a monotonic increase toward the graphite value. By carrying out calculations on high-symmetry giant fullerenes (Cn) in the size regime 100 < n <1000, we are able to demonstrate the beginnings of such a trend. In particular C540 is calculated to exhibit more than 10% of the π-electron ring current magnetic susceptibility of graphite on a per carbon basis. Endohedral chemical shifts are predicted to be invariant to cluster size, but subject to the quantum size effects seen in smaller fullerenes and metallic clusters. The fullerenes are different from the metallic clusters because the finite band gap in conjugated carbon compounds allows the diamagnetic term to dominate at large cluster size. The experimentally observed decrease in nanotube material diamagnetism with temperature is attributed to the increased importance of the Van Vleck term due to finite-temperature effects.