Autologous T Cells Transduced With Retroviral Vectors Expressing TCRs for Participant-specific Neoantigens in Patients With Hematologic Malignancies

* INCLUSION CRITERIA: Malignancy diagnosis requirements: -Eligible diagnoses include AML (acute myeloid leukemia), MDS (myelodysplastic syndrome), CMML(chronic myelomonocytic leukemia), CML (chronic myeloid leukemia), and T-ALL (T-acute lymphoblastic leukemia/lymphoma) meeting standard diagnostic criteria as described in the 5th edition World Health Organization Classification of Hematologic Tumors and/or the International Consensus Classification of Myeloid Neoplasms and Acute Leukemias. Multiple myeloma participants meeting International Working Group diagnostic criteria are eligible. These diagnostic criteria can be met at any time during the course of the participant s malignancy. Atypical CML is not an eligible diagnosis. NOTE: Pathology reports are acceptable to confirm eligibility. Malignancy mutation and HLA requirements: * Detection of at least one of the neoepitope-forming TP53 or RAS mutations that are listed in Table 3 in on the TruSight Oncology (TSO) 500 sequencing panel (NSR device) performed in the NCI Laboratory of Pathology is required. RAS mutations can be in NRAS, KRAS or HRAS as these oncogenes have the same amino acid sequence at the location of the targeted neoepitopes. A variant allele frequency (VAF) of at least 5% is required for a mutation to be eligible. This criterion can be met at any time within 60 days prior to apheresis regardless of treatment history during this 60-day period. DNA for sequencing comes from bone marrow. * Presence of the correct HLA type needed to present one of the targeted neoepitopes as shown in Table 3. HLA typing data from any time-point prior to apheresis can be used to meet this requirement. Table 3: Eligibility requirements for the targeted mutation and HLA type Targeted mutation - TP53 R175H; HLA Type - A\*02:01 Targeted mutation - TP53 Y220C; HLA Type - A\*02:01 Targeted mutation - TP53 R248W; HLA Type - A\*68:01 Targeted mutation - Ras G12V; HLA Type - A\*11:01 Targeted mutation - Ras G12D; HLA Type - A\*11:01 Targeted mutation - Ras G12D; HLA Type - C\*08:02 Targeted mutation - Ras G12V; HLA Type - C\*01:02 Malignancy burden requirements: * For AML and MDS, bone marrow myeloblast percentage must be \>=5% of nucleated cells in either bone marrow aspirate or biopsy. Myeloblasts can be defined by immunohistochemistry or by cytochemistry stains including but not limited to myeloperoxidase. * For T-ALL, bone marrow T-cell blast percentage must be \>=5% of nucleated cells in either bone marrow aspirate or biopsy. T cells can be defined by cytochemistry or immunohistochemistry or flow cytometry. * For multiple myeloma, plasma cells having a phenotype consistent with multiple myeloma must be detected at any frequency by multiparameter bone marrow flow cytometry or total plasma cells must be at least 6% on bone marrow core biopsy or bone marrow aspirate. * For CMML, bone marrow blast (including monocytic blast equivalent) percentage must be \>=6% of bone marrow nucleated cells by cytochemistry or immunohistochemistry of bone marrow aspirate or biopsy. * For CML measurable leukemia is defined as molecular detection of BCR-ABL1 at a ratio of \>1.0% to ABL1 or another housekeeping gene on The International Scale (IS) in either blood or bone marrow. Malignancy prior treatment and risk category criteria \- Participants with AML, MDS, CML, CMML, and T-ALL who have not had prior allogeneic hematopoietic stem cell transplantation (alloHSCT) must be unwilling or unable to undergo alloHSCT. NOTE: Unable to undergo alloHSCT could be due to lack of access to transplantation or not meeting transplant eligibility criteria at one or more transplant centers where the participant was evaluated by a transplant physician. * Participants with primary, secondary, or treatment-related AML that did not go into remission after induction therapy are eligible regardless of history of alloHSCT. * Myelodysplastic syndrome (MDS) * Participants with MDS must have had high or very high risk MDS as determined by IPSS-R or IPSS-M (https://mds-risk-model.com) at any time point. * Participants with MDS must have received previous treatment with at least one of the following: a hypomethylating agent, cytotoxic chemotherapy, or alloHSCT. Participants with primary or treatment-related MDS are eligible. * Participants with MDS/AML with mutated TP53 are eligible. * Participants with CMML must have had a CMML-specific prognostic scoring system-Molecular (CPSS-Mol) score of \>=2 (Intermediate-2 or High risk groups) at any time-point and must have received at least one line of previous systemic treatment, which could have been alloHSCT. * Chronic myeloid leukemia (CML) * Participants with chronic phase CML and a history of inadequate response to or intolerance of 3 or more tyrosine kinase inhibitors (TKIs) are eligible. * In addition, participants who have received at least one of bosutinib, dasatinib, or nilotinib in addition to either ponatinib or asciminib are eligible. Participants in accelerated phase or blast crisis are eligible if they have received at least one TKI. * Participants who have received a prior HSCT are eligible provided they have also received at least 2 TKIs and meet other eligibility criteria. * Participants with T-ALL must have T-ALL that did not go into CR with induction therapy or that relapsed. * Participants with relapsed AML who are unable to undergo alloHSCT and meet other eligibility requirements are eligible. * Multiple Myeloma * Participants with multiple myeloma must have received at least 3 different prior systemic treatment regimens for multiple myeloma. Participants must have prior exposure to an imid such as lenalidomide, a proteosome inhibitor, and a BCMA-targeting CAR T-cell therapy, such as monoclonal antibody, or bispecific antibody. * Multiple myeloma participants with a history of alloHSCT are eligible * Participants with multiple myeloma must also have measurable multiple myeloma defined by at least one of the criteria below: * Serum M-protein greater or equal to 1.0 g/dL. * Urine M-protein greater or equal to 200 mg/24 h. * Serum free light chain (FLC) assay: involved FLC level greater or equal to 10mg/dL (100 mg/L) provided serum FLC ratio is abnormal. * A biopsy-proven plasmacytoma at least 2.0 cm in largest dimension. * Bone marrow core biopsy with 30% or more plasma cells. Other inclusion criteria * Blast cells \<=1% of white blood cells as measured by CBC and differential before apheresis * Plasma cells \<=1% of white blood cells as measured by CBC and differential before apheresis * Participants must be willing to undergo intensive care unit care including mechanical ventilation if necessary * Participants must not have received systemic chemotherapy for at least 14 days prior to start of lymphodepleting chemotherapy or apheresis, and chemotherapy-related toxicities other than cytopenias must have recovered to grade 0 or grade 1 by the time of apheresis. The one exception is if necessary to control AML, CML, or CMML, hydroxyurea can be administered up to 7 days prior to apheresis. * Participants who have received alloHSCT must have received a transplant from either a fully matched sibling or 10/10 HLA-matched unrelated donor. * Recipients of alloHSCT must be at least 100 days post-transplant before the apheresis. * Subjects must be willing to be co-enrolled on NCI protocol 03C0277 and 09C0161. * Age must be \>=18 and \<= 75 years old * Clinical performance status of ECOG 0 or 1 * Participants must have adequate organ function as defined below: * Hemoglobin: \>=8 g/dL without red blood cell transfusions for 7 days prior to blood count check * Platelets: \>=45,000/mcL without transfusion support in the 7 days prior to the blood count check * Absolute neutrophil count: \>=850/mcL without exogenous growth factor administration within the 10 days prior to the blood count check * Total bilirubin: \<= 2.0 mg/dL. Except for participants with Gilbert s syndrome (who must have a total bilirubin \<3 mg/dL) * Alanine transaminase (ALT) and aspartate transaminase (AST): \<= to 3 times the upper limit of the institutional normal unless liver involvement by malignancy is demonstrated. If liver involvement with malignancy is detected, ALT and AST must be \<= 5 times the upper limit of normal * Serum Creatinine: \<= 1.5 mg/dL * Participants who have received prior genetically-engineered T-cell therapies are eligible if at least 180 days have elapsed between the date of previous T-cell infusion and apheresis. * Room air oxygen saturation must be 93% or greater * Women of child-bearing potential (WOCBP) 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. NOTE: IOCBP is defined as any person who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal. Men able to father children 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 these Men with partners of childbearing potential ask their partners to be on highly effective birth control (hormonal, intrauterine device \[IUD\], surgical sterilization). Men able to father a child must not freeze or donate sperm within the same period. * Nursing participants must be willing to discontinue breastfeeding from study treatment initiation through 4 months after the last dose of the study drug(s). * Hepatitis B surface antigen and hepatitis B core antibody tests must be negative. If either of these tests are positive, participants must have a negative blood PCR test for hepatitis B to enroll on the study. * Hepatitis C antibody test must be negative. If this test is positive, participants must have a negative blood PCR test for hepatitis C RNA to enroll on the study. * Cardiac ejection fraction of greater than or equal to 50% by echocardiography with no evidence of hemodynamically significant pericardial effusion as determined by an echocardiogram within 30 days prior to apheresis. * All participants must be willing to undergo mandatory bone marrow biopsy/ aspirates during the study. * Participants with a history of cigarette smoking of \>5 pack years, a history of pulmonary disease, a history of alloHSCT, or chronic pulmonary symptoms must undergo pulmonary function testing and have an FEV1 \>50% predicted and diffusing capacity for carbon monoxide \>= 60%. * Subjects who received a previous allogeneic HSCT must have no (grade 0) acute GVHD and no chronic GVHD or mild chronic GVHD as defined. --NOTE: Subjects with GVHD meeting the above criteria with local therapy (topical cutaneous steroids, inhaled steroids, and eye drops) will be eligible. * Potential participants must agree to stay within 1-hour drive of NIH clinical center from date of initial discharge until at least 14 days have elapsed since T cell infusion through the 14 day time period. * Ability of the participant to understand and the willingness to sign a written informed consent document. * Willing to sign a durable power of attorney. EXCLUSION CRITERIA: -For alloHSCT recipients only, subjects receiving any systemic immunosuppressive drugs including corticosteroids at doses of greater than 5 mg/day prednisone or equivalent within 28 days prior to apheresis. NOTE: Topical corticosteroid preparations applied to the skin such as solutions, creams, and ointments are allowed. Inhaled corticosteroids are allowed, and corticosteroid eye drops are allowed. * Corticosteroids given for any indication at doses greater than 5 mg/day of prednisone or equivalent within 14 days before either apheresis or start of protocol chemotherapy. * Participants with MDS/Myeloproliferative neoplasia overlap syndromes are not eligible. * Participants with acute promyelocytic leukemia are not eligible. * Participants who received a mis-matched sibling or haploidentical transplant are not eligible. * Tumor masses \>=10 cm in largest diameter * Positive beta Human chorionic gonadotropin (beta-HCG) serum or urine pregnancy test in IOCBP performed at screening. * Human T-cell lymphotropic virus type 1/ 2 (HTLV-1/II) positive * HIV infection, as measured by seropositivity for HIV antibody. * Participants that require urgent therapy due to tumor mass effects on vital organ or tumor lysis syndrome. * Any significant illness that, in the opinion of the principal investigator, may impair the participant s tolerance of the study treatment as evaluated by medical history, physical exam, assess for hepatosplenomegaly, and chemistry laboratory evaluations. * Participants with a history of a previous malignancy are ineligible if the malignancy has not been in complete remission for at least 2 years or if the previous malignancy required treatment with surgery, radiation, or chemotherapy, including maintenance hormonal therapy, in the past 2 years. Exceptions to this requirement are participants who have had successful resection of the following types of skin cancer: nonmetastatic basal cell carcinoma or squamous cell carcinoma or stage 0 melanoma. * Suspected or confirmed active uncontrolled infections defined as fevers of \>38 degrees within the past 24 hours without a known non-infectious source or participants requiring intravenous antibiotics when intravenous antibiotics have been administered for less than 72 hours. * Acti...