AFPᶜ³³²T in Advanced HCC

Key Inclusion Criteria: 1. Subject is ≥ 18 years and ≤ 75 years of age and has voluntarily agreed to participate by giving written informed consent in accordance with ICH GCP Guidelines and applicable local regulations. 2. Histologically confirmed HCC, not amenable to transplant, resection. Subjects may undergo loco-regional therapy after enrollment but not at time of lymphodepletion. Or Histologically confirmed diagnosis of another AFP expressing tumor (e.g. cholangiocarcinoma). 3. Measurable disease according to RECIST 1.1 criteria prior to lymphodepletion. 4. Progressive disease following or intolerant of or refuses standard of care systemic therapy prior to lymphodepletion. 5. Positive for any A\*02:01 P group allele. 6. a) Group 1, 2, 3, (HCC) Subjects will be eligible for enrollment if they meet either one of these AFP expression criteria: * AFP expression of ≥1+ in ≥20% of tumor cells by immunohistochemistry and their non-cancerous liver tissue has ≤5% cells stained for AFP at any intensity by immunohistochemistry. * Serum AFP levels of ≥100ng/mL and their non-cancerous liver tissue has ≤5% cells stained for AFP at any intensity by immunohistochemistry. 6\. b) Group 4 (other AFP expressing tumor types) Subjects will be eligible for enrollment if they meet either one of these AFP expression criteria: o Serum AFP levels of ≥100ng/mL and their non-cancerous liver tissue (if applicable) has ≤5% cells stained for AFP at any intensity by immunohistochemistry. 7\. Life expectancy of \> 4 months 8. Child-Pugh score ≤ 6 9. Eastern Cooperative Oncology Group (ECOG) 0-1 10. Subject must have adequate organ function as defined in the protocol. Key Exclusion Criteria: 1. Positive for any of the HLA-A\*02 allele other than HLA-A\*02:01 P Group, HLA-A\*02:03 P group or null alleles or positive for the following alleles: HLA-C\*04:04 or HLA-B\*51:03. 2. Prior liver transplant 3. Received the following prior to leukapheresis: 1. Cytotoxic chemotherapy, immune therapy and biological therapy within 3 weeks 2. Corticosteroids or any other immunosuppressive therapy within 2 weeks. NOTE: Use of inhaled or topical steroids is not exclusionary 3. Sorafenib/Regorafenib/Lenvatinib within 1 week 4. Cabozantinib within 2 weeks 5. Duration of treatment free intervals for any other anti-cancer therapies must be discussed with Adaptimmune Study Physician. 4. Received the following prior to lymphodepleting chemotherapy : 1. Cytotoxic chemotherapy or loco-regional therapy within 3 weeks, liver directed radiation therapy within three months. 2. Corticosteroids or any other immunosuppressive therapy within 2 weeks. NOTE: Use of inhaled or topical steroids is not exclusionary 3. Bone/soft tissue directed palliative radiotherapy within 4 weeks. 4. Investigational treatment or clinical trial within 4 weeks. 5. Sorafenib/Regorafenib/Lenvatinib within 1 week. 6. Cabozantinib within 2 weeks 7. Prior cancer-directed immunotherapy within 4 weeks, including monoclonal antibodies against PD-1 receptor or ligand. 8. Use of an experimental vaccine within 2 months in the absence of tumor response. The subject should be excluded if their disease is responding to an experimental vaccine given within 6 months 9. Any previous gene therapy using an integrated vector 10. Duration of treatment free intervals for any other anti-cancer therapies must be discussed with Adaptimmune Study Physician. 5. Toxicity persisting from previous anti-cancer therapy of ≥ Grade 2 (except for non-clinically significant toxicities, e.g., alopecia, vitiligo). Subjects with Grade 2 toxicities that are deemed stable or irreversible (e.g. peripheral neuropathy) can be enrolled on a case-by-case basis with prior consultation and agreement with the Sponsor Study Physician. 6. Major surgery within 4 weeks prior to lymphodepletion; subjects should have been fully recovered from any surgical related toxicities. 7. Bleeding ≥ Grade 2 in the past 3 months prior to lymphodepletion 8. Therapeutic anticoagulation from lymphodepletion until platelet count recovery (prophylactic heparin allowed) 9. Clinically or radiographically detectable ascites (beyond trace/rim of ascites) or ascites requiring medication 10. Clinically detectable hepatic encephalopathy or hepatic encephalopathy requiring medication 11. Active viral hepatitis Subjects positive for hepatitis B surface antigen not on antiviral treatment/prophylaxis or subjects with detectable hepatitis B DNA 1. Subjects with resolved (surface antigen negative, core antibody positive) or chronic stable (surface antigen positive) hepatitis B on antiviral treatment/prophylaxis with DNA levels of ≤100 IU/mL are allowed with HBV DNA monitoring after treatment 2. Subjects with hepatitis C allowed provided they meet all other eligibility criteria 12. Positive serology for HIV 13. Positive serology for HTLV 1 or 2 14. History of chronic or recurrent (within the last year) severe autoimmune or immune mediated disease requiring steroids or other immunosuppressive treatments. Subjects with history of idiopathic autoimmune hepatitis are excluded. Subjects who experienced hepatitis during treatment with check point inhibiting antibodies are not excluded. 15. Subject has brain metastases. 16. Other active malignancy besides HCC or other eligible AFP expressing tumor types within 3 years. 17. Electrocardiogram (ECG) showing clinically significant abnormality at Screening or showing a QTc interval ≥450 msec in males and ≥470 msec in females (≥480 msec for subjects with Bundle Branch Block (BBB) over consecutive ECGs). 18. Bacterial or opportunistic infection within 3 months of treatment (upper respiratory infection and uncomplicated urinary tract infection allowed) 19. Uncontrolled intercurrent illness considered by the Investigator to add appreciable risk to study participation, including but not limited to: 1. Clinically significant cardiac disease defined by CHF New York Heart Association (NYHA) \> Class 1; uncontrolled clinically significant arrhythmia in last 6 months; Acute Coronary Syndrome (ACS) (angina or myocardial infarction) in last 6 months. 2. Oxygen dependent lung disease. 3. Clinically significant psychiatric illness/social situations that would limit compliance with study requirements. 4. History of stroke or central nervous system bleeding; transient ischemic attack (TIA) or reversible ischemic neurologic deficit (RIND) in last 6 months. 20. Pregnant or breastfeeding 21. Alcohol or illicit drug dependency 22. Known contraindication to cyclophosphamide, fludarabine, mesna, G-CSF or other agents associated with study treatment.