Cord Blood Transplant in Adults With Blood Cancers

Inclusion Criteria: * I. Acute myelogenous leukemia (AML): * Complete first remission (CR1) at high risk for relapse such as any of the following: * Known prior diagnosis of myelodysplasia (MDS) or myeloproliferative disorder (MPD). * Therapy-related AML. * Presence of extramedullary leukemia at diagnosis. * Requirement for 2 or more inductions to achieve CR1. * Intermediate or high ELN2017 genetic risk AML. * Any patient unable to tolerate consolidation chemotherapy as would have been deemed appropriate by the treating physician. * Other high-risk features not defined above. * Complete second remission (CR2) or greater (CR2+). * Patients in morphologic remission with persistent cytogenetic, flow cytometric, or molecular aberrations are eligible II. Acute lymphoblastic leukemia (ALL): * Complete first remission (CR1) at high risk for relapse such as any of the following: * Presence of any high-risk cytogenetic abnormalities such as t(9;22), t(1;19), t(4;11) or other MLL rearrangements (11q23) or other high-risk molecular abnormality. * Failure to achieve MRD- complete remission after induction therapy. * Persistence or recurrence of minimal residual disease on therapy. * Any patient unable to tolerate consolidation and/or maintenance chemotherapy as would have been deemed appropriate by the treating physician. * Other high-risk features not defined above. * Complete second remission (CR2) or greater (CR2+). Note: ALL with less than 5% blasts at time of transplant but persistent cytogenetic, flow cytometric or molecular aberrations are eligible. III. Other acute leukemias: Acute leukemias of ambiguous lineage or mixed phenotype with less than 5% blasts. Leukemias in morphologic remission with persistent cytogenetic, flow cytometric or molecular aberrations are eligible. IV. Myelodysplastic Syndromes (MDS) and Myeloproliferative Disorders (MPD) other than myelofibrosis: * International prognostic scoring system (IPSS) risk score of INT-2 or high risk at the time of diagnosis. * Any IPSS risk category if life-threatening cytopenia(s) exists. * Any IPSS risk category with karyotype or genomic changes that indicate high risk for progression to acute myelogenous leukemia. * MDS/MPD overlap syndromes without myelofibrosis. * MDS/ MPD patients must have less than 10% bone marrow myeloblasts and ANC \> 0.2 (growth factor supported if necessary) at transplant work-up. V. Non-Hodgkin lymphoma (NHL) at high-risk of relapse or progression if not in remission: Eligible patients with aggressive histologies (such as, but not limited to, diffuse large B-cell NHL, mantle cell NHL, and T-cell histologies) in CR by PET/CT imaging. o Eligible patients with indolent B-cell NHL (such as, but not limited to, follicular, small cell or marginal zone NHL) will have 2 nd or subsequent progression with PR or CR by PET/CT imaging. VI. Blastic plasmacytoid dendritic cell neoplasm (BPDCN) in morphologic remission. Organ Function and Performance Status Criteria: * Karnofsky score equal or greater than 80% (See Appendix B; inpatient Leukemia service transfers without discharge are acceptable provided patient has equivalent KPS as if were outpatient). * Calculated creatinine clearance \> 70 ml/min. * Bilirubin \< 1.5 mg/dL (unless benign congenital hyperbilirubinemia or hemolysis). * ALT \< 3 x upper limit of normal (ULN). * Pulmonary function: Spirometry (FVC and FEV1) and corrected DLCO) \> 60% predicted. * Left ventricular ejection fraction (MOD-bp)\> 50%. * Albumin \> 3.0. * Hematopoietic Cell Transplantation Comorbidity index (HCT-CI) ≤5. Graft criteria: Two CB units will be selected according to current MSKCC CB unit selection algorithm. High resolution 8-allele HLA typing and recipient HLA antibody profile will be performed. Unit selection will occur based on HLA-match, total nucleated cell (TNC) and CD34+ cell dose adjusted per patient body weight. The bank of origin will also be considered. Donor specific HLA antibodies, if present, will also be taken into consideration and may influence the selection of the graft. * Each CB unit must be at least 3/8 HLA-matched to the patient considering high-resolution 8-allele HLA typing. \[Taken from the Cord Blood Summary\] * Each CB unit will be required to have a cryopreserved TNC dose of at least 1.5 x 10\^7 TNC/ recipient body weight (TNC/ kg). \[Taken from the Cord Blood Summary\] * Each CB unit will be required to have a cryopreserved CD34+ cell dose of at least 1.5 x 10\^5 CD34+ cells/ recipient body weight (CD34+ cells/kg). \[Taken from the Cord Blood Summary\] * A minimum of one unit will be reserved as a backup graft. \[Taken from the Cord Blood Summary\] * Each CB unit will be required to be cryopreserved in standard cryovolume (24-27 ml/s per unit or per bag if unit in two bags) and be red blood cell depleted. \[Taken from the Cord Blood Summary\] Exclusion Criteria: * Diagnosis of myelofibrosis or other malignancy with moderate-severe bone marrow fibrosis. * Patients with persistent with CNS involvement in CSF or CNS disease at time of screening * Prior checkpoint inhibitors/ blockade in the last 12 months. * Two prior stem cell transplants of any kind. * One prior autologous stem cell transplant within the preceding 12 months. * Prior allogeneic transplantation. * Prior involved field radiation therapy that would preclude safe delivery of 400cGy TBI in the opinion of Radiation Oncology. * Active and uncontrolled infection at time of transplantation. * HIV infection. * Seropositivity for HTLV-1. * Inadequate performance status/ organ function. * Pregnancy or breast feeding. * Patient or guardian unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, long-term follow-up, and research tests.