A method of producing a purified argon product (174, 195) in a cryogenic air separation unit integrated with a pressure swing adsorption system, the method comprising the steps of: separating argon from an oxygen-argon containing stream within a lower pressure column of the cryogenic air separation unit, the separation of the argon from the oxygen-argon containing stream to produce an impure argon stream (97B, 98B) having between about and 4% and 25% of oxygen impurities; warming the impure argon stream via indirect heat exchange in an argon recovery heat exchanger (152, 173); compressing the warmed, impure argon stream; purifying the warmed, pressurized impure argon stream by introducing the warmed, pressurized impure argon stream into a first stage pressure swing adsorption system having at least two adsorbent beds (210, 211) each having an adsorbent configured for adsorbing the oxygen impurities in the warmed, pressurized impure argon stream and each adsorbent bed in the first stage pressure swing adsorption system configured to produce a high purity argon stream and a first waste argon-oxygen containing stream; further compressing the first waste argon-oxygen containing stream; introducing the first waste argon-oxygen containing stream into a second stage pressure swing adsorption system having at least two adsorbent beds (220, 221) each having an adsorbent configured for adsorbing the oxygen impurities in the first waste argon-oxygen containing stream and each adsorbent bed in the second stage pressure swing adsorption system configured to produce a moderate purity argon stream and a second waste argon-oxygen containing stream (172A, 222); recycling the moderate purity argon stream to a location upstream of the first stage pressure swing adsorption system; wherein the impure argon stream is warmed via indirect heat exchange in the argon recovery heat exchanger against the high purity argon stream (170, 174), the second waste argon-oxygen containing stream, and a balancing air stream of purified, compressed air; and recycling the second waste argon-oxygen containing stream to an argon rectification column (56, 90) in the cryogenic air separation unit. separating argon from an oxygen-argon containing stream within a lower pressure column of the cryogenic air separation unit, the separation of the argon from the oxygen-argon containing stream to produce an impure argon stream (97B, 98B) having between about and 4% and 25% of oxygen impurities; warming the impure argon stream via indirect heat exchange in an argon recovery heat exchanger (152, 173); compressing the warmed, impure argon stream; purifying the warmed, pressurized impure argon stream by introducing the warmed, pressurized impure argon stream into a first stage pressure swing adsorption system having at least two adsorbent beds (210, 211) each having an adsorbent configured for adsorbing the oxygen impurities in the warmed, pressurized impure argon stream and each adsorbent bed in the first stage pressure swing adsorption system configured to produce a high purity argon stream and a first waste argon-oxygen containing stream; further compressing the first waste argon-oxygen containing stream; introducing the first waste argon-oxygen containing stream into a second stage pressure swing adsorption system having at least two adsorbent beds (220, 221) each having an adsorbent configured for adsorbing the oxygen impurities in the first waste argon-oxygen containing stream and each adsorbent bed in the second stage pressure swing adsorption system configured to produce a moderate purity argon stream and a second waste argon-oxygen containing stream (172A, 222); recycling the moderate purity argon stream to a location upstream of the first stage pressure swing adsorption system; wherein the impure argon stream is warmed via indirect heat exchange in the argon recovery heat exchanger against the high purity argon stream (170, 174), the second waste argon-oxygen containing stream, and a balancing air stream of purified, compressed air; and recycling the second waste argon-oxygen containing stream to an argon rectification column (56, 90) in the cryogenic air separation unit.