This first time in human study is a single-arm study of genetically engineered MAGE A10ᶜ⁷⁹⁶T cells in HLA-A*02:01 and/or HLA-A*02:06 positive subjects with advanced (Stage IIIb or IV) NSCLC. Subjects will be screened for general health, performance status and disease stage. Following screening, subjects meeting all eligibility will undergo a large-volume leukapheresis to obtain cells for the manufacture of autologous MAGE A10ᶜ⁷⁹⁶TCR bearing T cells. When the MAGE A10ᶜ⁷⁹⁶T cells are available, subjects will receive lymphodepleting chemotherapy with cyclophosphamide on Days -7 and -6 or cyclophosphamide and fludarabine on Days -7, Day -6, and Day -5 followed by the T cell infusion on Day 1.