Autologous T-cells Genetically Engineered to Express Receptors Reactive Against KRAS Mutations in Conjunction With a Vaccine Directed Against These Antigens in Participants With Metastatic Cancer

* INCLUSION CRITERIA:
* Participants with an appropriate HLA match for available Surgery Branch KRAS TCRs with evaluable metastatic solid cancer (e.g., gastrointestinal, genitourinary, breast, ovarian, non-small cell lung cancer (NSCLC) and other solid cancers) with known KRAS G12V or G12D mutation.
* Confirmation of diagnosis of cancer by the NCI Laboratory of Pathology.
* Refractory to standard systemic therapy. Specifically:

  * Participants with metastatic colorectal cancer must have received oxaliplatin and/or irinotecan.
  * Participants with breast and ovarian cancer must have received at least two systemic 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 other solid tumors must have received at least one prior line of systemic treatment or have declined standard treatment.
  * Participants with three (3) or fewer brain metastases that are \< 1 cm in diameter each 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 \>= 18 years and \<= 72 years.
* Clinical performance status of ECOG 0 or 1.
* Individuals of child-bearing potential (IOCBP) must agree to use highly effective contraception (hormonal, intrauterine device \[IUD, abstinence, surgical sterilization starting at the time of study entry, for the duration of study therapy, and 12 months after the last dose of combined chemotherapy

Participants who can father a child must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and for 4 months after the last dose of combined chemotherapy. We also will recommend participants that can father children with partners of childbearing potential to ask their partners to be on highly effective birth control (hormonal, intrauterine device (IUD), surgical sterilization).

NOTE: IOCBP is defined as any person who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal.

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.

* Participants must have serology results as follows:

  * Seronegative for HIV antibody.
  * 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.
* Adequate organ and marrow function as defined below:

  --Hematology:
  * ANC \> 1000/mm\^3 without growth factor support
  * WBC \>= 2500/mm\^3
  * Platelet count (Bullet) 80,000/mm3
  * Hemoglobin \> 8.0 g/dL. Subjects may be transfused to reach this cut-off.
* Chemistry:
* Serum ALT/AST \<= 5.0 x ULN
* Serum creatinine \<= 1.6 mg/dL
* Total bilirubin \<= 2.0 mg/dL, except in participants with Gilbert s Syndrome, who must have a total bilirubin \< 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.

* For participants with NSCLC or lung metastases, 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.
* Participants 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.
* Any form of secondary immunosuppression.
* Active systemic infections requiring anti-infective treatment, coagulation disorders, or any other active or uncompensated major medical illnesses.
* For participants with NSCLC or lung metastases, 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.
* 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, aldesleukin or vaccines.
* Clinically significant participant history which in the judgment of the Principal Investigator (PI) would compromise the participants ability to tolerate high-dose aldesleukin.
* History of coronary revascularization or ischemic symptoms.
* For select participants with a clinical history prompting cardiac evaluation: last known LVEF \<= 45%.
* For select participants with a clinical history prompting pulmonary evaluation: known FEV1 \<= 50% predicted.