A new formulation of the current within the London approximation allows the calculation of ring currents in topologically complex molecules. Application of this theory to C60 demonstrates the existence of remarkable pi-electron ring currents. Paramagnetic currents, in size comparable to the ones in benzene, flow within the pentagons, whereas weaker diamagnetic currents flow all around the C60 molecule. The overall vanishing ring-current magnetic susceptibility results from a cancellation of diamagnetic and paramagnetic contributions. The presence of ring currents significantly affects chemical shifts as measured in nuclear magnetic resonance experiments. In contrast to the magnetic susceptibility, which is a property of the molecule as a whole, chemical shifts are sensitive to the local magnetic field and the effect of ring currents does not vanish.