NOVELTY - Producing ceria-zirconia solid solution crystal fine particles comprises: obtaining a melt containing 5-50 mol.% of zirconium oxide and cerium oxide, 10-60 mol.% of metal oxide compound (I) and 25-70 mol.% of boron oxide; quenching the melt to obtain an amorphous material; heating the amorphous material at 550-1000 degrees C to precipitate ceria-zirconia solid solution crystals in the amorphous material; and separating components other than the ceria-zirconia solid solution crystals from the obtained precipitates.    USE - The fine particles are useful as an oxidation-reduction catalyst, as a material for a co-catalyst of a three-way catalyst for treatment of exhaust gas discharged from automobiles, and as a material for fuel cells and as a polishing material.    ADVANTAGE - The process produces ceria-zirconia solid solution crystal fine particles with high crystallinity, excellent heat resistance, small particle size and high specific surface area. The ceria-zirconia solid solution crystal fine particles are easily produced and exhibits excellent uniformity in composition and particle size. The fine particles undergo insignificant change in the particle size and the specific surface area even in high-temperature use.    DETAILED DESCRIPTION - Producing ceria-zirconia solid solution crystal fine particles comprises: obtaining a melt containing 5-50 mol.% of zirconium oxide and cerium oxide, 10-60 mol.% of metal oxide compound of formula (RO) (I) and 25-70 mol.% of boron oxide; quenching the melt to obtain an amorphous material; heating the amorphous material at 550-1000 degrees C to precipitate ceria-zirconia solid solution crystals in the amorphous material; and separating components other than the ceria-zirconia solid solution crystals from the obtained precipitates.    R = Mg, Ca (preferred), Sr or Ba.    An INDEPENDENT CLAIM is included for ceria-zirconia solid solution crystal fine particles comprising 70-99.9 mol.% of zirconium oxide and cerium oxide, 0.01-20 mol.% of (I) and 0.01-10 mol.% of boron oxide.