We consider circular currents in molecular wires with loop substructures studied within simple tight-binding models. Previous studies of this issue have focused on specific molecular structures. Here we address several general issues. First, we consider the quantitative definition of a circular current and adopt a definition that identified the circular component of a loop current as the sole source of the magnetic field induced in the loop. The latter may be associated with the field at the loop center, with the magnetic moment associated with this field, or with the total magnetic flux threading the loop. We show that all three definitions yield an identical measure of the loop current. Secondly, we study dephasing effects on the loop current and the associated induced magnetic field. Finally, we consider circular currents in several molecular structures: benzene, azulene, naphthalene and anthracene and show that circular currents occur generically in such structures and can be, in certain voltage ranges, considerably larger than the net current through the molecule, and are furthermore quite persistent to normal thermal dephasing.