Inclusion Criteria, MAC RECIPIENTS:

1. Age 18 to \< 66 years (chemotherapy-based conditioning) or \< 61 years (TBI-based conditioning) at the time of signing informed consent
2. Patient or legally authorized representative has the ability to provide informed consent according to the applicable regulatory and institutional requirements
3. Stated willingness to comply with all study procedures and availability for the duration of the study
4. Planned MAC regimen (see Table 8 in Section 7.4 for allowed MAC regimens)
5. Available partially HLA-MMUD (4/8-7/8 at HLA-A, -B, -C, and -DRB1 is required) with age 16-35
6. Product planned for infusion is MMUD T-cell replete PBSC as allograft
7. HCT-CI \< 5 (Appendix H - Hematopoietic Cell Transplant Comorbidity Index Scoring). The presence of prior malignancy will not be used to calculate HCT-CI for this trial, to allow for the inclusion of patients with secondary or therapy-related AML or MDS.
8. One of the following diagnoses:

1. AML, ALL, or other acute leukemia in first remission or beyond with ≤ 5% marrow blasts and no circulating blasts or evidence of extramedullary disease. Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning.
2. Patients with MDS with no circulating blasts and with \< 10% blasts in the bone marrow (higher blast percentage allowed in MDS due to lack of differences in outcomes with \< 5% vs 5-10% blasts in MDS). Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning.
9. Cardiac function: Left ventricular ejection fraction ≥ 45% based on most recent echocardiogram or multi-gated acquisition scan (MUGA) results
10. Estimated creatinine clearance ≥ 45mL/min calculated by equation
11. Pulmonary function: diffusing capacity of the lungs for carbon monoxide (DLCO) corrected for hemoglobin ≥ 50% and forced expiratory volume in first second (FEV1) predicted ≥ 50% based on most recent PFT results
12. Liver function acceptable per local institutional guidelines
13. KPS of ≥ 70% (Appendix I - Performance Status)

Inclusion Criteria, RIC/NMA RECIPIENTS:

1. Age ≥ 18 years at the time of signing informed consent
2. Patient or legally authorized representative has the ability to provide informed consent according to the applicable regulatory and local institutional requirements
3. Stated willingness to comply with all study procedures and availability for the duration of the study
4. Planned NMA/RIC regimen (see
5. Table 9 in Section 7.4 for allowed NMA/RIC regimens)
6. Available partially HLA-MMUD (4/8-7/8 at HLA-A, -B, -C, and -DRB1 is required) with age 16-35
7. Product planned for infusion is MMUD T-cell replete PBSC allograft
8. One of the following diagnoses:

1. Patients with acute leukemia or chronic myeloid leukemia (CML) with no circulating blasts, no evidence of extramedullary disease, and with \< 5% blasts in the bone marrow. Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning.
2. Patients with MDS with no circulating blasts and with \< 10% blasts in the bone marrow (higher blast percentage allowed in MDS due to lack of differences in outcomes with \< 5% vs 5-10% blasts in MDS.) Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning.
3. Patients with chronic lymphocytic leukemia (CLL) or other leukemias (including prolymphocytic leukemia) with chemosensitive disease at time of transplantation.
4. Higher-risk chronic myelomonocytic leukemia (CMML) according to CMML-specific prognostic scoring system or high-risk MDS/myeloproliferative neoplasms (MPN) not otherwise specified are eligible, provided there is no evidence of high-grade bone marrow fibrosis or massive splenomegaly at the time of enrollment.
5. Patients with lymphoma with chemosensitive disease at the time of transplantation.
6. Patients with primary myelofibrosis or myelofibrosis secondary to essential thrombocythemia, polycythemia vera or MDS with grade 4 fibrosis.
9. Cardiac function: Left ventricular ejection fraction ≥ 40% based on most recent echocardiogram or MUGA results with no clinical evidence of heart failure
10. Estimated creatinine clearance ≥ 45mL/min calculated by equation
11. Pulmonary function: DLCO corrected for hemoglobin ≥ 50% and FEV1 predicted ≥ 50% based on most recent PFT results
12. Liver function acceptable per local institutional guidelines
13. KPS of ≥ 60% (Appendix I - Performance Status)

Exclusion Criteria:

1. Suitable HLA-matched related or 8/8 high-resolution matched unrelated donor available
2. Subject unwilling or unable to give informed consent, or unable to comply with the protocol including required follow-up and testing
3. Subjects with a prior allogeneic transplant
4. Subjects with an autologous transplant within the past 3 months
5. Subjects who are breastfeeding or pregnant
6. Uncontrolled bacterial, viral or fungal infection at the time of the transplant preparative regimen
7. Concurrent enrollment on a GVHD prevention clinical trial
8. Subjects who undergo desensitization to reduce anti-donor HLA antibody levels prior to transplant
9. Patients who are HIV-positive with persistently positive viral load. HIV-infected patients on effective anti-retroviral therapy (ART) with undetectable viral load within 6 months are eligible for this trial. Patients with well-controlled HIV are eligible provided resistance panels are negative, the patient is compliant with ART, and their disease remains well controlled.