A process for making cerium dioxide nanoparticles containing at least one transition metal (M) utilizes a suspension of cerium hydroxide nanoparticles prepared by mechanical shearing of an aqueous mixture containing an oxidant in an amount effective to enable oxidation of cerous ion to ceric ion, thereby forming a product stream that contains transition metal-containing cerium dioxide nanoparticles, Ce1-xMxO2, wherein “x” has a value from about 0.3 to about 0.8. The nanoparticles thus obtained have a cubic fluorite structure, a mean hydrodynamic diameter in the range of about 1 nm to about 10 nm, and a geometric diameter of less than about 4 nm. The transition metal-containing crystalline cerium dioxide nanoparticles can be used to prepare a dispersion of the particles in a nonpolar medium.