A Basket Study of Customized Autologous TCR-T Cell Therapies in Patients With Locally Advanced (Unresectable) or Metastatic Solid Tumors

Inclusion Criteria: 1. Must be at least 18 years. 2. Locally advanced (unresectable) or metastatic solid tumor for which there are no available curative treatment options, after failure of the standard of care systemic therapies for that particular indication. 3. Solid tumors, including but not limited to non-nasopharyngeal head and neck cancer, non-small cell lung cancer, cutaneous melanoma, cervical cancer, ovarian cancer, anal cancer and genital cancers. Other tumor types may be permitted if approved by TScan. 4. Participants must express one of the following HLA types, as assessed by a qualified genomics assay in screening study TSCAN-003: HLA-B\*07:02, HLA-A\*01:01, HLA-C\*07:02 and/or HLA-A\*02:01 5. Tumor must express one or more of the following: MAGE-A1, MAGE-A4, MAGE-C2, PRAME and HPV16 assessed in the last 8 months in screening study TSCAN-003 (NCT05812027). 6. Eastern Cooperative Oncology Group (ECOG) Performance status 0-1 at screening. 7. Participants must be able to understand and be willing to give informed consent; decision-impaired adults may consent with their legally authorized representative. 8. At least 1 measurable lesion per modified Response Evaluation Criteria in Solid Tumors (RECIST) v1.1. 9. Adequate bone marrow and organ function. Exclusion Criteria: 1. Medical or psychological conditions that would make the participant unsuitable candidate for cell therapy at the discretion of the PI. 2. History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, cardiac arrhythmia requiring antiarrhythmic or procedure, or other clinically significant cardiac disease within 12 months of enrollment 3. Have a history of ASTCT Grade 4 CRS, Grade 3 or greater ICANS, or Grade 3 or greater IECHS. Participants with a history of lower grade CRS, ICANS, or IECHS may be eligible, pending review and approval by the Medical Monitor. 4. History of stroke or transient ischemic attack (TIA) within 6 months of enrollment 5. Systemic corticosteroid therapy \>10 mg of prednisone daily or equivalent within 7 days of enrollment. 6. History of severe hypersensitivity to fludarabine or cyclophosphamide or study product excipients including human serum albumin, Cryostor (DMSO or Dextran 40), or Plasma-Lyte. 7. Untreated or symptomatic central nervous system (CNS) metastases or cytology proven carcinomatous meningitis. 8. Concurrent receipt of another anti-cancer therapy. Have a history of acute mental status changes of unknown etiology within 6 months prior to enrollment, or any neurological or neurodegenerative disorder (e.g., Parkinson disease, Huntington disease, uncontrolled seizure disorder) that may increase the risk for or confound the assessment of neurotoxicity. 9. Presence of fungal, bacterial, viral, or other infection requiring anti-microbials for management. 10. Tumors that have HLA LOH using a central lab clinical trial assay of HLAs addressed by the monotherapy and/or T-Plex combination TCR-Ts in the protocol and have no available TCR-T options for intact HLAs in the participant's tumor. 11. Participants who regularly require supplemental oxygen.