A method and apparatus for combustion of a fuel in which a first-stage fuel and a first-stage oxidant are introduced into a combustion chamber and ignited, forming a primary combustion zone. At least about 5% of the total heat output produced by combustion of the first-stage fuel and the first-stage oxidant is removed from the primary combustion zone, forming cooled first-stage combustion products. A portion of the cooled first-stage combustion products from a downstream region of the primary combustion zone is recirculated to an upstream region of primary combustion zone. A second-stage fuel is introduced into the combustion chamber downstream of the primary combustion zone and ignited, forming a secondary combustion zone. At least about 5% of the heat from the secondary combustion zone is removed. In accordance with one embodiment, a third-stage oxidant is introduced into the combustion chamber downstream of the secondary combustion zone, forming a tertiary combustion zone.