Abstract: Method and apparatus for catalytically oxidating grease, fats, oils and/or other hydrocarbons in fumes rising from a source thereof. In particular, grease and odor-laden air streams, i.e. hydrocarbon-laden air streams, emanating from a source such as a cooking grill or the like are heated to a temperature at which passage over a catalyst causes flameless oxidation of the organic material in the air stream at a temperature below combustion temperature, the organic material being converted essentially to carbon dioxide and water, with release of at least some heat energy. The oxidized and heated air stream is ducted in countercurrent relation or in identically parallel direction, to the relatively cooler hydrocarbon-laden air stream which is being ducted toward the catalyst, heat exchange from the oxidized air stream to the hydrocarbon-laden air stream thereby occurring to at least partially heat the hydrocarbon-laden air stream.