Carbon nanotubes with their cylindrical topology belong to the class of systems where the Aharonov-Bohm effect can manifest. We study one of its manifestations, i.e. the persistent currents and magnetic moments which they induce. We explore the dependence of these magnetic moments on the size of the nanotube, its chirality and on their chemical potential, referring to recently performed experiments. In multiwall nanotubes the interaction between persistent currents induced in individual shells may lead to collective phenomena such as spontaneous flux or flux trapping. The influence of some of the possible corrections to the standard tight-binding dispersion relation for graphene is also discussed.