Donor Stem Cell Transplantation Using α/β+ T-lymphocyte Depleted Grafts From HLA Mismatched Donors

Subject Inclusion Criteria: * Patients with any of the following hematologic malignancies who are considered to be eligible for allogeneic transplantation: * Acute lymphoid leukemia (ALL) in first complete remission (CR1) with high risk for relapse including: * Detectable minimal residual disease by either multicolor flow cytometry or by genomic assay after initial induction therapy * t(9;22) or detected BCR-ABL1 translocation by genomic methodologies * BCR-ABL1-Like B-ALL \[23\] including mutations of IKZF1 or CRLF2 * Translocations or mutations involving 11q23 (MLL) gene. * Hypodiploid karyotype * Deletion of 9p * Loss of 17p or TP53 mutation * T-lymphocyte lineage antigen expression (T-ALL) * Prior CNS or other extramedullary involvement * WBC count ≥ 100,000 cells/μL at diagnosis * Acute biphenotypic or bilineal leukemia in CR1 * Acute myeloid leukemia (AML) in CR1 with * Detectable minimal residual disease (MRD) by either multicolor flow cytometry or by genomic assay after initial induction therapy * In the absence of MRD any intermediate or high risk features according to the European LeukemiaNet 2017 guidelines indlucing: * Mutated FL T3-ITD or FL T3-TKD * Cytogenetic abnormalities not classified as favorable * Cytogenetic abnormalities associated with myelodysplastic syndrome including abnormalities of chromosome 5, 7, or 17p * Complex karyotype or monosomal karyotype * t(9;11)(p21.1;q23.3); MLL-KMT2A or other rearrangements of KMT2A * t(9;11); BCR-ABL1 * Inversions or translocations of chromosome 3 * T(6;9)(p23;q34.1); DEK-NUP214 * Somatic mutation of RUNX1, ASX1 or TP53 * Extramedullary involvement * WBC count ≥100,000 cells/μL at diagnosis * Relapsed acute leukemia with ≤ 5% blasts in the bone marrow prior to transplantation (i.e. CR2 or greater). * Myelodysplastic syndrome, myeloproliferative neoplasms, or MDS/MPN overlap syndrome with ≤ 10% blasts and at least one of the following: * Revised International Prognostic Scoring System risk score of INT, HIGH, or VERY HIGH at the time of transplant evaluation. * Life-threatening cytopenias * Karyotype or genomic changes that indicate high risk for progression to acute myelogenous leukemia, including abnormalities of chromosome 7 or 3, mutations of TP53, or complex or monosomal karyotype. * Therapy related disease or disease evolving from other malignant processes. * Chronic myelomonocytic leukemia (CMML) with ≤ 10% blasts prior to transplantation. * Chronic myeloid leukemia (CML) meeting one of the following criteria: * Failed or are intolerant to BCR-ABL tyrosine kinase inhibitors. * CML with BCR-ABL mutation consistent with poor response to tyrosine kinase inhibition (e.g. T351I mutation). * CML with accelerated or blast phase with \<10% blasts after therapy. * Chronic lymphocytic leukemia (CLL) with high risk disease as defined by the EBMT consensus criteria * Hodgkin lymphoma meeting both of the following criteria: * Responding to therapy prior to enrollment * Relapse after autologous bone marrow transplant or are ineligible for autologous bone marrow transplant. °Non-Hodgkin lymphoma meeting both of the following criteria: * Responding to therapy prior to enrollment. * Relapse after prior autologous bone marrow transplant or are ineligible for autologous bone marrow transplant. * Patients aged from birth through 65 years old are eligible. * Patients must have Karnofsky/Lanksy performance status ≥70%. * Cardiac left ventricular ejection fraction ≥50% at rest. * Serum bilirubin ≤ 2 mg/dL. Patients with Gilbert's disease or ongoing hemolytic anemia are acceptable if the direct bilirubin is ≤ 2 mg/dL. * AST and ALT ≤ 2.5 x ULN unless thought to be disease related * Estimated or measured creatinine clearance \> 50 mL/min/1.73 m\^2 body surface area. * Adult patients and pediatric patients capable of performing pulmonary function studies must have hemoglobin adjusted pulmonary DLCO ≥50% of predicted. Subject Exclusion Criteria: * Persons with a HLA matched sibling donor or a 8/8 allele level HLA-matched unrelated donor. * Female patients who are pregnant or breast-feeding. * Persons with an infection that is not responding to antimicrobial therapy. * Persons who are seropositive for HIV. * Persons with active/detectable central nervous system malignancy. * Persons who do not meet the age and organ function criteria specified above. * Presence of psychiatric or neurologic disease, or lack of social support that limits the patient's ability to comply with the treatment protocol including supportive care, followup, and research tests. * Prior allogeneic hematopoietic cell transplantation are ineligible. * Patients with history of other malignancy within 5 years of study therapy are ineligible with the following exceptions: Low grade prostate cancer (Gleason's ≤6) treated with curative intent, breast ductal carcinoma in situ treated with curative intent, or nonmelanomatous skin carcinomas. Donor Inclusion and Exclusion Criteria: * Partially HLA-matched unrelated volunteers (allele level matched at 6-7 of 8 HLA loci: -A, -B, -C, and -DRB1) are eligible. * Related, haploidentical donors are eligible. * Able to provide informed consent to the donation process * Meet standard criteria for donor collection as defined by the National Marrow Donor Program Guidelines.