An improvement of a combustion furnace for a boiler in which it is possible to improve and uniformize the heat collection ratio of the furnace of the boiler. Moreover, this invention easily controls the temperature of superheated steam while the heating surface area is kept constant without altering the boiler construction and heat balance, and provides a zone temperature control to a boiler. The boiler according to the present invention is provided with a radiant heat transfer portion (15) in which at least one or more heat accumulating type burner systems are disposed in which a pair of burners (2, 3) for suppyling air for combustion and exhausting combustion gas through a heat accumulator (5) are caused to burn in an alternate fashion while combustion gas is discharged through the heat accumulator (5) from one of the burners which is not burning, so that excess heat energy that has not been consumed in the radiation heat transfer portion (15) is recovered when it is exhausted via the heat accumulator (5) of the other burners. In addition, in the boiler of the present invention, a plurality of heat accumulating type burner systems (1) are disposed in a direction in which water for the boiler flows, and the pair of heat accumulating type burners (2, 3) of the respective burner systems (1) are disposed in a direction that intersects the direction in which water for the boiler flows, whereby a temperature zone is formed by controlling combustion for each heat accumulating type burner system (1), thereby making it possible to control the distribution of the furnace heat collection ratio. Furthermore, the boiler of the present invention has the radiation heat transfer portion (15) in which at least one heat accumulating type burner system is provided, and is provided with a superheater (17) in which saturated steam is superheated by extracting a part of combustion gas from the radiation heat transfer portion (15), the combustion gas being exhausted directly out of the combustion chamber without passing through the burners (2, 3).