This invention is a combustion process having a series of stages in which the fuel is combusted stepwise using specific catalysts (desirably palladium-bearing catalysts in the first two zones and metal and oxygen-bearing catalysts in the hot catalytic zone) and catalytic structures and, optionally, a final homogeneous combustion zone. The choice of catalysts and the use of specific structures, including those employing integral heat exchange, results in a catalyst support which is stable due to its comparatively low temperature and yet the product combustion gas is at a temperature suitable for use in a gas turbine, furnace, boiler, or the like, but has low NO.sub.x content.