  Current densities are induced in the electronic structure of molecules when
they are exposed to external magnetic fields. Aromatic molecular rings sustain
net diatropic ring currents, whereas the net ring current in antiaromatic
molecular rings is paratropic and flows in the opposite, non-classical
direction. We present computational methods and protocols to calculate, analyse
and visualise magnetically induced current densities in molecules. Calculated
ring-current strengths are used for quantifying the degree of aromaticity. The
methods have been demonstrated by investigating ring-current strengths and the
degree of aromaticity of aromatic, antiaromatic and non-aromatic six-membered
hydrocarbon rings. Current-density pathways and ring-current strengths of
aromatic and antiaromatic porphyrinoids and other polycyclic molecules have
been studied. The aromaticity and current density of M\"obius-twisted molecules
has been investigated to find the dependence on the twist and the spatial
deformation of the molecular ring. Current densities of fullerene, gaudiene and
toroidal carbon nanotubes have also been studied.
