NOVELTY - Composite oxide comprises 60-98 mass% of a cerium-containing element in terms of oxide, the cerium-containing element consisting of cerium and at least one element selected from rare earth metal elements other than cerium and including yttrium, zirconium, and silicon at 85:15 to 100:0 by mass in terms of oxides; 1-20 mass% of an alkaline earth metal element in terms of oxide; and 1-20 mass% of aluminum in terms of aluminum oxide (Al2O3).    USE - The composite oxide is useful for catalyst for purifying exhaust gas (claimed), which reduces or eliminates nitrogen oxide (NOx). It can further be used in functional ceramics, solid electrolyte for fuel cells, and abrasive.    ADVANTAGE - In the composite oxide and catalyst, the oxide has excellent heat resistance, including that a large specific surface area is maintained even when the composite oxide is used in a high temperature environment, and that, even after calcination at 800 degrees C for 2 hours, no AECeO3 phase is detected and increase in CeO2 crystallite size is inhibited.    DETAILED DESCRIPTION - Composite oxide comprises 60-98 mass% of a cerium-containing element in terms of oxide, the cerium-containing element consisting of cerium and at least one element selected from rare earth metal elements other than cerium and including yttrium, zirconium, and silicon at 85:15 to 100:0 by mass in terms of oxides; 1-20 mass% of an alkaline earth metal element in terms of oxide; and 1-20 mass% of aluminum in terms of aluminum oxide (Al2O3); where the composite oxide has properties of exhibiting a specific surface area of not smaller than 40 m2/g as measured by the Brunauer-Emmett-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 cerium oxide (CeO2) crystallite size in (111) plane of not larger than 15 nm, as determined by X-ray diffraction after calcination at 800 degrees C for 2 hours. INDEPENDENT CLAIMS are:    (1) a method for producing a composite oxide comprising: (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 lower than 60 degrees C to obtain a cerium suspension, (C) adding at least precursors of an alkaline earth metal oxide and aluminum 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 lower 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) a catalyst for purifying exhaust gas comprising the composite oxide.