Weekly ADI-PEG 20 will be cohort dose escalated (18, 27 and 36 mg/m2), with pemetrexed 500 mg/m2 and cisplatin 75 mg/m2 both given every 3 weeks. Subjects may receive a maximum of 6, 3-week cycles of ADIPemCis for a total of 18 weeks of treatment. Subjects with NSCLC may receive 4 to 6, 3-week cycles as per local institutional policy. Those subjects completing ADIPemCis treatment may continue on ADI-PEG 20 monotherapy if they have SD or better. Subjects with NSCLC may continue to receive pemetrexed as per local institutional policy along with ADI-PEG 20 and/or continue on ADI-PEG 20 monotherapy after pemetrexed is also discontinued.