Administering Peripheral Blood Lymphocytes Transduced With a CD70-Binding Chimeric Antigen Receptor to People With CD70 Expressing Cancers

* INCLUSION CRITERIA: * For Phase I: Evaluable, unresectable cancer expressing CD70 as assessed by immunohistochemistry of resected tissue (greater than or equal to 2+ CD70 positive on greater than or equal to 50% of cancer cells, or greater than or equal to 1+ CD70 positive on greater than or equal to 75% of cancer cells). * For Phase II: Measurable (per RECIST v1.1 criteria), unresectable cancer expressing CD70 as assessed by immunohistochemistry of resected tissue (greater than or equal to 2+ CD70 positive on greater than or equal to 50% of cancer cells, or greater than or equal to 1+ CD70 positive on greater than or equal to 75% of cancer cells). * Confirmation of the diagnosis of cancer by the NCI Laboratory of Pathology. * Patients must have previously received at least one standard therapy for their cancer (if available) and have been either non-responders (progressive disease) or have recurred. * Patients with 3 or fewer brain metastases that are less than or equal to 1 cm in diameter and asymptomatic are eligible. Lesions that have been treated with stereotactic radiosurgery must be clinically stable for 1 month after treatment for the patient to be eligible. Patients with surgically resected brain metastases are eligible. * Age greater than or equal to 18 years and less than or equal to 72 years. * Clinical performance status of ECOG 0 or 1 * Patients of both sexes must be willing to practice birth control from the time of enrollment on this study and for 12 months after the last dose of combined chemotherapy for women and for four months after treatment for men. * Women of child-bearing potential must be willing to undergo a pregnancy test prior to the start of treatment because of the potentially dangerous effects of the treatment on the fetus. NOTE: Certain malignancies may secrete hormones that produce false positive pregnancy tests. Serial blood testing (e.g. HCG measurements) and/ or ultrasound may be performed for clarification. -Serology --Seronegative for HIV antibody. (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who are HIV seropositive may have decreased immune-competence and thus be less responsive to the experimental treatment and more susceptible to its toxicities.) * Seronegative for hepatitis B antigen, and seronegative for hepatitis C antibody. If hepatitis C antibody test is positive, then patient must be tested for the presence of antigen by RT-PCR and be HCV RNA negative. -Hematology * ANC greater than 1000/mm(3) without the support of filgrastim * WBC greater than or equal to 2500/mm(3) * Platelet count greater than or equal to 80,000/mm(3) * Hemoglobin \> 8.0 g/dL. Subjects may be transfused to reach this cut-off. -Chemistry * Serum ALT/AST less than or equal to 5.0 times ULN * Serum creatinine less than or equal to 1.6 mg/dL * Total bilirubin less than or equal to 2.0 mg/dL, except in patients with Gilbert s Syndrome who must have a total bilirubin less than 3.0 mg/dL. * Patients must have completed any prior systemic therapy at the time of enrollment. Note: Patients may have undergone minor surgical procedures or limited field radiotherapy within the four weeks prior to enrollment, as long as related major organ toxicities have recovered to grade 1 or less. * Ability of subject to understand and the willingness to sign a written informed consent document. * Willing to sign a durable power of attorney. * Subjects must be co-enrolled on the NCI-SB cell harvest protocol 03-C-0277 (Cell Harvest and Preparation for Surgery Branch Adoptive Cell Therapy Protocols). EXCLUSION CRITERIA: * Women of child-bearing potential who are pregnant or breastfeeding because of the potentially dangerous effects of the treatment on the fetus or infant. * Concurrent systemic steroid therapy. * Active systemic infections requiring anti-infective treatment, coagulation disorders, or any other active or uncompensated major medical illnesses. * Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease). * History of hematopoietic autoimmune disease or any autoimmune disease requiring immunosuppressive measures. * Concurrent opportunistic infections (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who have decreased immune-competence may be less responsive to the experimental treatment and more susceptible to its toxicities). * History of severe immediate hypersensitivity reaction to cyclophosphamide, fludarabine, or aldesleukin. * History of coronary revascularization or ischemic symptoms. * For select patients with a clinical history prompting cardiac evaluation: last known LVEF less than or equal to 45%. * For select patients with a clinical history prompting pulmonary evaluation: known FEV1 less than or equal to 50% predicted. * Patients who are receiving any other investigational agents.