Room-temperature ferromagnetism has been observed in the nanoparticles (7 - 30 nm dia) of nonmagnetic oxides such as CeO2, Al2O3, ZnO, In2O3 and SnO2. The saturated magnetic moments in CeO_2 and Al_2O_3 nanoparticles are comparable to those observed in transition metal doped wide band semiconducting oxides. The other oxide nanoparticles show somewhat lower values of magnetization but with a clear hysteretic behavior. Conversely, the bulk samples obtained by sintering the nanoparticles at high temperatures in air or oxygen became diamagnetic. As there were no magnetic impurities present, we assume that the origin of ferromagnetism may be due to the exchange interactions between localized electron spin moments resulting from oxygen vacancies at the surfaces of nanoparticles. We suggest that ferromagnetism may be a universal characteristic of nanopartilces of metal oxides