This investigation demonstrates a strong dependence of the magnetism of CeO₂ nanoparticles on the concentration of oxygen vacancies. A strong magnetic signal is over for a narrow range (0.1 < Ce³⁺/Ce < 0.2) of Ce³⁺ concentrations in the nanoparticles. Previous studies have determined that most of the vacancies are at the surface of the particle and, given particles of size 3-10 nm, the strong magnetic signal corresponds to an even narrower range of Ce³⁺ content at the surface (0.40 < Ce³⁺_surface/Ce_surface < 0.45). X-ray magnetic circular dichroism (XMCD) measurement reveals that electrons in Ce bear magnetic moments while oxygen atoms do not respond magnetically indicating that the bounded magnetic polarons are more important to magnetism than is oxygen-mediated exchange.