In the late 1920s Niels Bohr propagated the idea that the magnetic moment of a free electron could not be observed. This derived from the idea that the spin degree of freedom characterized the electron only when it is bound in an atom. This view initiated a lively discussion, which involved many of the most prominent theoreticians of the time. The independent existence of the electron spin became an issue of principle. In particular it was deemed that quantum effects would destroy the separated classical trajectories in a Stern-- Gerlach type of experiment. We review these discussions and some later developments. Quantum effects do prove to be essential, but they do not overwhelm the magnetic effects of spin. In addition to these arguments, it has been possible experimentally to determine the electron g-factor with high accuracy in electromagnetic traps. In fact no principle seems to prevent the observation of the magnetic moment of the free electron.