Brief Summary: Background: Human papillomavirus (HPV) can cause cervical, throat, anal, and genital cancers. Cancers caused by HPV have a HPV protein called E7 inside of their cells. In this new therapy, researchers take a person s blood, remove certain white blood cells, and insert genes that make them to target cancer cells that have the E7 protein. The genetically changed cells, called E7 TCR cells, are then given back to the person to fight the cancer. Researchers want to see if this can help people. Objective: To determine a safe dose and efficacy of E7 TCR cells and whether these cells can help patients. Eligibility: Adults ages 18 and older with an HPV-16-associated cancer, including cervical, vulvar, vaginal, penile, anal, or oropharyngeal. Design: Participants will list all their medicines. Participants will have many screening tests, including imaging procedures, heart and lung tests, and lab tests. They will have a large catheter inserted into a vein. Participants will have leukapheresis. Blood will be removed through a needle in the arm. A machine separates the white blood cells. The rest of the blood is returned through a needle in the other arm. The cells will be changed in the lab. Participants will stay in the hospital. Over several days, they will get: Chemotherapy drugs E7 TCR cells Shots or injections to stimulate the cells Participants will be monitored in the hospital up to 12 days. They will get support medicine and have blood and lab tests. Participants will have a clinic visit about 40 days after cell infusion. They will have a physical exam, blood work, scans, and maybe x-rays. Participants will have many follow-up visits with the same procedures. At some visits, they may undergo leukapheresis. Participants will be followed for 15 years.

Detailed Description: Background: * Metastatic or refractory/recurrent human papillomavirus (HPV)-16+ cancers (cervical, vulvar, vaginal, penile, anal, and oropharyngeal cancers) are incurable and poorly palliated by standard therapies. * HPV-16+ cancers constitutively express the HPV-16 E7 oncoprotein, which is absent from healthy human tissues. * Administration of T cell receptor (TCR) gene engineered T cells can induce objective tumor responses in certain malignancies including HPV-16+ cancers. * T cells genetically engineered with a TCR targeting HPV-16 E7 (E7 TCR) display specific reactivity against HLA-A2+, HPV-16+ target cells. Objectives: Phase I Primary Objective - To determine a safe dose for E7 TCR cells plus aldesleukin for the treatment of metastatic HPV-16+ cancers. Phase II Primary Objective -To determine safety and efficacy of E7 TCR cells plus aldesleukin for the treatment of metastatic HPV-16+ cancers. Eligibility: * Patients greater than or equal to 18 years old with metastatic or refractory/recurrent HPV-16+ cancer. * Prior first line systemic therapy is required unless the patient declines standard treatment. * Patients must be HLA-A\*02:01-positive. Design: * This is a phase I/II clinical trial that will test the safety and efficacy of E7 TCR cells. * All patients will receive a non-myeloablative lymphocyte-depleting preparative regimen of cyclophosphamide and fludarabine followed by a single infusion of E7 TCR cells. Cell infusion will be followed by high-dose aldesleukin. * Re-enrollment will be allowed for a small number of subjects.