Control of magnetism by applied voltage is desirable for spintronics applications. Finding a suitable material remains an elusive goal, with only a few candidates found so far. Graphene is one of them and attracts interest because of its weak spin-orbit interaction, the ability to control electronic properties by the electric field effect and the possibility to introduce paramagnetic centres such as vacancies and adatoms. Here we show that the magnetism of adatoms in graphene is itinerant and can be controlled by doping, so that magnetic moments are switched on and off. The much-discussed vacancy magnetism is found to have a dual origin, with two approximately equal contributions; one from itinerant magnetism and the other from dangling bonds. Our work suggests that graphene's spin transport can be controlled by the field effect, similar to its electronic and optical properties, and that spin diffusion can be significantly enhanced above a certain carrier density.