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Cerium oxide nanoparticles and nanoporous thin films exhibiting room temperature magnetism have been synthesized by two different methods. The magnetization curves for both particles and films are almost anhysteretic; they fit the empirical expression M ≈ Mstanh(H/H0) , where Ms and H0 are the saturation magnetization and extrapolated saturation field respectively. It is estimated that about 0.1-0.5% of the volume of the CeO2 samples are magnetic. This is consistent with models where the magnetism is associated with grain boundaries, surfaces or interfaces which occupy only a small fraction of the entire sample volume. The effect in the electrodeposited films depends on film morphology, which is controlled by the electrochemical variables. The results are discussed in terms of current models for d0 magnetism.