On the magnetic criterion, an aromatic system is one that sustains a diatropic ring current. Once this definition is accepted, calculation of current density induced in a molecule by an external magnetic field gives a direct way of determining aromaticity from ab initio calculation. The ipsocentric treatment of molecular magnetic response, in which each point is the origin for the local current-density, offers an accurate, economical and conceptually transparent approach to the calculation of ring currents. The distribution of origin uniquely leads to physically non-redundant orbital contributions· that can be used to interpret (or predict) currents in terms of orbital symmetries, energies and nodal character. Diatropic and paratropic currents obey translational and rotational selection rules, respectively, leading to a 'spectroscopic', 'frontier-orbital' theory of ring currents and magnetic aromaticity. This thesis reports investigations of the nature and origin of ring currents in 1t-, (j- and homo-aromatic systems. Rules developed for annulenes extend to heterocycles and account for retention of ring-current aromaticity in perfluorobenzenes in which argon atoms are progressively inserted in the CF bonds. Currents in benzenoid and nonbenzenoid polycyclic aromatic hydrocarbons give varied patterns which can be rationalised in terms of Kekule structures and Pauling bond orders. These concepts are used to explain rim-and-hub currents in circulenes and design molecules with fully paramagnetic 1t currents, verified by ab initio calculation. Studies of heteropolycycles with carbocyclic cores demonstrate the aromaticity of mellitic trianhydride, the non-aromaticity of 'quasi-rings' closed by hydrogen- and lithium-bonds, and rationalise the magnetic properties of the new 'sulflowers' . Calculations on model rings of silicon atoms with tangential p-p bonding show the possibility of (j ring currents governed by modified versions of the ipsocentric 1t selection rules. Saturated cyclopropane also supports (j ring current. The homotropylium cation illustrates the association of a ring current of modified topology with homoaromaticity.