Cerium oxide abrasive and method of polishing substrates

A cerium oxide abrasive slurry having, dispersed in a medium, cerium oxide particles whose primary particles have a median diameter of from 30 nm to 250 nm, a maximum particle diameter of 600 nm or smaller, and a specific surface area of from 7 to 45 m.sup.2 /g, and slurry particles have a median diameter of from 150 nm to 600 nm. The cerium oxide particles have structural parameter Y, representing an isotropic microstrain obtained by an X-ray Rietvelt method (with RIETAN-94), of from 0.01 to 0.70, and structural parameter X, representing a primary particle diameter obtained by an X-ray Rietvelt method (with RIETAN-94), of from 0.08 to 0.3. The cerium oxide abrasive slurry is made by a method of obtaining particles by firing at a temperature of from 600.degree. C. to 900.degree. C. and then pulverizing, then dispersing the resulting cerium oxide particles in a medium.