NOVELTY - A composite oxide comprises: a cerium-containing element in terms of oxide, where the cerium-containing element consists of cerium and at least one element selected from rare earth metal elements other than cerium and including yttrium, zirconium, and aluminum at 85:15 to 100:0 mass% in terms of oxides; an alkaline earth metal element (1-30 mass%) in terms of oxide; and silicon (1-20 mass%) in terms of SiO2.    USE - As a catalyst for purifying exhaust gas (claimed); and also used for functional ceramics, solid electrolyte for fuel cells, and abrasive.    ADVANTAGE - The composite oxide exhibit a specific surface area of not less than 60 m2/g as measured by the Brunauer, Emmet, and Teller (BET) method after calcination at 800 degrees C for 2 hours, and having no AECeO3 phase (where AE stands for an alkaline earth metal element) and having a CeO2 crystallite size in (111) plane of not greater than 15 nm, as determined by X-ray diffraction after calcination at 800 degrees C for 2 hours. The composite oxide has a specific surface area of not less than 80 m2/g as measured by the BET method after calcination at 800 degrees C for 2 hours. The composite oxide has excellent heat resistance. The method allows easy production of the composite oxide.    DETAILED DESCRIPTION - INDEPENDENT CLAIMS are included for the following:    (1) producing a composite oxide involves: (A) providing a cerium solution not less than 90 mole% of which cerium ions are tetravalent, (B) heating and holding the cerium solution obtained from step (A) up to and at not less than 60 degrees C to obtain a cerium suspension, (C) adding at least precursors of an alkaline earth metal oxide and silicon oxide to the cerium suspension obtained from step (B) to obtain a suspension, (D) heating and holding the suspension obtained from step (C) up to and at not less than 100 degrees C, (E) adding a first precipitant to the suspension obtained from step (D) to precipitate elements other than the alkaline earth metal element, (F) adding a second precipitant to obtain a precipitate containing the alkaline earth metal element, and (G) calcining the precipitate obtained from step (F); and    (2) new catalyst for purifying exhaust gas comprising the composite oxide.