A composite oxide includes CeO.sub.2, ZrO.sub.2 and a metallic oxide being free from reacting with CeO.sub.2 and ZrO.sub.2 at 700.degree. C. or more, preferably at 900.degree. C. or more and further preferably at 1,000.degree. C. or more. The composite oxide has a regulatory oriented phase, such as a pyrochlore phase, etc., in which at least a part of Ce cations and Zr cations are oriented regularly. The composite oxide makes a catalytic support. The CeO.sub.2--ZrO.sub.2 composite oxide and the reaction-free metallic oxide make barriers each other which suppress the granular growth when the composite oxide is subjected to high-temperature heat in a reducing heat treatment to form the regulatory oriented phase. The regulatory oriented phase improves the oxygen storage-and-release capability of the catalytic support. Thus, it is possible to simultaneously attain a large a specific surface area and a high oxygen storage-and-release capability.