Abstract: Flue gases containing crackable and combustible volatiles e.g. from a wood-burning stove are passed through a catalytic combustor having two spaced catalytic units each having small longitudinally extending catalytic flow passages. Air supplied into a chamber between the two units supports after-combustion of combustible volatiles in the downstream catalytic unit. Some pre-cracking and combustion takes place in the upstream catalytic unit, which serves as a gas collimator and disallows any flow of air from the intermediate chamber into the fire box of the stove, whereby combustion in the fire box can be closely controlled to increase production of combustible volatiles. Each unit may have flow passages which decrease in cross-sectional size and increase in length toward the central portion of the unit, thus encouraging uniformity of flow of the flue gases and of the catalytic reaction over the entire cross section of each unit.