Administration of Autologous T-Cells Genetically Engineered to Express T-Cell Receptors Reactive Against Neoantigens in People With Metastatic Cancer

* INCLUSION CRITERIA: * Metastatic, solid cancer that can be measured, and falls into one of five cohorts: (1) gastrointestinal and genitourinary cancers; (2) breast, ovarian, and other solid cancers; (3) non-small cell lung cancer (NSCLC); (4) endocrine tumors including neuroendocrine tumors; and, (5) multiple myeloma that includes measurable solid tumors (plasmacytomas). Participants with multiple myeloma are potentially eligible only if they have measurable multiple myeloma as defined in Section 16.7 after plasmacytoma resection. Note: NSCLC includes but is not limited to squamous cell carcinoma, adenosquamous carcinoma, or adenocarcinomas. * Documented diagnosis of cancer. * Refractory to approved standard systemic therapy. Specifically: * Participants with metastatic colorectal cancer must have received oxaliplatin or irinotecan. * Participants with breast and ovarian cancer must be refractory to first- line treatment and refractory to or have refused second-line treatments. * Participants with NSCLC must have received at least one platinum-based chemotherapy regimen and at least one FDA-approved targeted treatment (when appropriate). * Participants with endocrine tumors including neuroendocrine tumors must be refractory to first-line therapy (e.g., lanreotide, octreotide) and must be refractory or have refused second-line treatments such as everolimus, sunitinib, or 177 Lu-Dotatate, if indicated. * Participants with multiple myeloma must have received at least four prior lines of therapy that included at least one exposure to an immunomodulatory drug such as lenalidomide, a proteosome inhibitor, an anti-CD38 antibody treatment, and an autologous stem cell transplant. * Participants with three (3) or fewer brain metastases that are \< 1 cm in diameter and asymptomatic are eligible. Lesions that have been treated with stereotactic radiosurgery must be clinically stable for one month after treatment for the participant to be eligible. Participants 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. * Participants of both sexes must be willing to practice birth control from the time of enrollment on this study and for and 12 months after the last dose of combined chemotherapy for individuals of child-bearing potential (IOCBP) and four months after treatment for participants who can father a child. * Individuals 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. Participants 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 participant must be tested for the presence of antigen by RT-PCR and be HCV RNA negative. * Hematology: * ANC \> 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 x 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 participants with Gilbert's Syndrome, who must have a total bilirubin less than or equal to 3.0 mg/dL. * Participants must have completed any prior systemic therapy at the time of enrollment. Note: Participants 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. In addition, participants with multiple myeloma may receive bridging therapy during the time between study enrollment and start of study therapy. This may be necessary due to the long time needed for cell production on this study. After bridging therapy and within 14 days of protocol treatment start, participants with multiple myeloma must still have measurable multiple myeloma. * For Cohort 3: More than two weeks must have elapsed since any prior palliation for major bronchial occlusion or bleeding at the time the patient receives the preparative regimen, and patient s toxicities must have recovered to a 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 protocol 03-C-0277. EXCLUSION CRITERIA: * Participants who are pregnant or nursing 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. * For Cohort 3: Any major bronchial occlusion or bleeding not amenable to palliation. * Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease and AIDS). * History of major organ autoimmune disease. * For Arm 2: Grade 3 or 4 major organ irAEs following treatment with anti-PD-1/PD-L1, including but not limited to myocarditis and pneumonitis. Note: Participants with grade 3 or 4 major organ irAEs may be enrolled on Arm 1 if all other eligibility criteria are met. * Concurrent opportunistic infections (The experimental treatment being evaluated in this protocol depends on an intact immune system. Participants 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. * For Cohorts 1, 2, 4. or 5: Clinically significant participant history which in the judgment of the Principal Investigator (PI) would compromise the participants ability to tolerate high-dose aldesleukin. Note: At the discretion of the PI, participants enrolled in Cohort 3 may receive low-dose aldesleukin. * History of coronary revascularization or ischemic symptoms. * For select participants with a clinical history prompting cardiac evaluation: last known LVEF less than or equal to 45%. * For select participants with a clinical history prompting pulmonary evaluation: known FEV1 less than or equal to 50% predicted. * Participants who are receiving any other investigational agents.