A burner having main air ports for issuing a secondary air at a relatively high flow velocity, and auxiliary air ports for issuing a secondary air at a relatively low flow velocity located upstream of the main air ports with respect to a stream of a gaseous fuel or a preformed mixture of fuel and primary air in which the fuel is enriched over and above a flammability limit. The secondary air issuing from auxiliary air ports may have its flow velocity reduced by utilizing speed reducting compartments formed in a combustion chamber or recesses formed in walls of the combustion chamber. Flame stabilizing plates may be provided near the auxiliary air ports to extend substantially perpendicularly to the stream of the gaseous fuel or the preformed mixture of fuel and primary air. The main air ports may be in the form of a plurality of slits formed continuously in side-by-side relation in a secondary air passage located in the middle of and along a passage for the gaseous fuel or the preformed mixture of fuel and primary air. The slits each have an upper end portion in the form of an arc or an arcuate polygon at the inner surface. The secondary air passage may be formed on its upstream side with increased thickness portions in which slits are formed and contiguous with the secondary air issuing slits. The passage for the gaseous fuel or the preformed mixture of fuel and primary air may be formed with offset portions to widen the passage.