A distributed-origin coupled Hartree-Fock method is used to compute and map the pi and total current densities induced by a magnetic field in the planar monocyclic molecules benzene (C6H6), borazine (B3N3H6), boroxine (B3O3H3), s-triazine (C3N3H3), the cyclopentadienyl anion (C5H5-), and the tropylium cation (C7H7+). The maps show that s-triazine, the cyclopentadienyl anion, and the tropylium cation have delocalized pi ring currents similar to that in benzene whereas the pi currents are localized on the nitrogens in borazine and on the oxygens in boroxine. The computed magnetic susceptibilities show trends that follow those observed for the ring currents and provide measures of the "aromaticity" of the molecules. Current density maps of the symmetric trifluorides of benzene (C6H3F3) and triazine (C3N3F3) and of perfluorobenzene (C6F6) show independent localized circulations of nr charge on the fluorines with a weak compression of the region of circulation in the ring.