This is an open label clinical trial. Patients with the HLA-A201, HLA-A205, and/or HLA-A206 allele and whose tumor expresses the NY-ESO-1 tumor antigen will be eligible to receive NY-ESO-1ᶜ²⁵⁹T. The trial is conducted entirely with outpatient procedures; however, patients may be hospitalized for the cytoreductive chemotherapy at the discretion of the investigator. Upon enrollment, patients will undergo leukapheresis for T cell collection, and their cells will be genetically engineered and expanded ex vivo. Seven days prior to receiving T-cells patients will undergo a cyclophosphamide conditioning regimen to potentiate the immunotherapy. The cell product will be infused as a single infusion (Day 0, typically Monday) to mitigate risks associated with unanticipated infusion reactions. Patients will be followed daily for the first week, weekly until 4 weeks, 8 weeks, and 12 weeks and then at 6 months and every 3 months until disease progression. Patients will undergo disease monitoring by MRI/CT scan (as appropriate for disease) at baseline, day 28, 8 weeks and 12 weeks, and months 6, and every 3 months until progression. Tumor biopsies will be taken at baseline, Week 8, and upon progression. In patients who have progressive disease following initial infusion but whose tumors continue to express NY-ESO-1, these patients may be eligible for a second infusion with redirected T cells. At disease progression, the interventional portion of the protocol ends and long term follow-up (LTFU) begins, in accordance with FDA regulations. LTFU occurs semiannually for up to 5 years post infusion and then annually thereafter for up to 15 years.