An oxygen scavenger composition includes a block co-polymer having at least one block including a polyester condensation polymer segment having cycloalkenyl group(s) or functionality directly or indirectly bonded to the polymer chain of said block, and having at least one second block including a polymer segment of a thermoplastic, film forming material. The block copolymer has been found to act as an oxygen scavenger agent under both ambient and refrigerated conditions, to be compatible with conventional film forming packaging materials, and to provide compositions exhibiting low tack which can be readily formed and processed using conventional film forming equipment. A packaging material, such as a film or laminated product suitable for packaging applications, can include the oxygen scavenger composition.