Double Cord Versus Haploidentical (BMT CTN 1101)

Inclusion Criteria: * Patients 18 to 70 years old * Patients must have available both: a)One or more potential related mismatched donors (biologic parent(s) or siblings (full or half) or children). At least low resolution DNA based human leukocyte antigen (HLA) typing at HLA-A, -B, and -DRB1 for potential haploidentical sibling donors is required. b)At least two potential umbilical cord blood units identified. Each unit must have a minimum of 1.5 x 10\^7/kg pre-cryopreserved total nucleated cell dose. For non-red blood cell depleted units, the minimum pre-cryopreserved total nucleated cell dose of each unit must be at least 2.0 x 10\^7/kg. Units must be HLA matched at a minimum of 4/6 to the recipient at HLA-A, HLA-B (at low resolution using DNA based typing) and HLA-DRB1 (at high resolution using DNA based typing). Confirmatory typing is not required for randomization. * Acute Lymphoblastic Leukemia (ALL) in first complete remission (CR1) that is NOT considered favorable-risk as defined by the presence of at least one of the following: Adverse cytogenetics such as t(9;22), t(1;19), t(4;11), other Mixed Lineage Leukemia (MLL) rearrangements; White blood cell counts of greater than 30,000/mcL (B-ALL) or greater than 100,000/mcL (T-ALL)at diagnosis; Recipient age older than 30 years at diagnosis; Time to CR greater than 4 weeks * Acute Myelogeneous Leukemia (AML) in CR1 that is NOT considered as favorable-risk. Favorable risk is defined as having one of the following: t(8.21) without CKIT mutation, inv(16) without CKIT mutation or t(16;16), normal karyotype with mutated NPM1 and not FLT-ITD, normal karyotype with double mutated CEBPA, Acute promyelocytic leukemia (APL) in first molecular remission at end of consolidation * Acute Leukemias in 2nd or subsequent CR * Biphenotypic/Undifferentiated/Prolymphocytic Leukemias in first or subsequent CR, adult T-cell leukemia/lymphoma in first or subsequent CR * Burkitt's lymphoma: second or subsequent CR * Lymphoma fulfilling the following criteria: Chemotherapy-sensitive (at least stable disease lymphomas that have failed at least 1 prior regimen of multi-agent chemotherapy and are INELIGIBLE for an autologous transplant. Patients with chronic lymphocytic leukemia (CLL) are not eligible regardless of disease status. * Performance status: Karnofsky score greater than or equal to 70%. Additional Patient Inclusion Criteria for Conditioning: * Patients with Adequate Physical Function as Measured by: a. Cardiac: Left ventricular ejection fraction at rest must be greater than or equal to 40%, or shortening fraction less than 25%; b. Hepatic: Bilirubin less than or equal to 2.5 mg/dL, except for patients with Gilbert's syndrome or hemolysis. Alanine aminotransferase (ALT), aspartate aminotransferase (AST), and Alkaline Phosphatase less than 5 x upper limit of normal; c. Renal: Serum creatinine within normal range, or if serum creatinine outside normal range, then renal function (measured or estimated creatinine clearance or GFR)greater than 40 mL/min/1.73m\^; d. Pulmonary: Diffusing capacity of the lung for carbon monoxide (DLCO) (corrected for hemoglobin), forced expiratory volume in one second (FEV1), and forced vital capacity (FVC) greater than 50% predicted; * Additional Patient Inclusion Criteria for Patients Assigned to Haploidentical BM Arm: Patients must be HLA typed at high resolution using DNA based typing at the following HLA-loci: HLA-A, -B, -C and DRB1 and have available a related haploidentical BM donor with 2, 3, or 4 HLA-mismatches. A unidirectional mismatch in either the graft versus host or host versus graft direction is considered a mismatch. The donor and recipient must be HLA identical for at least one antigen (using high resolution DNA based typing) at the following genetic loci: HLA-A, HLA-B, HLA-C, and HLA-DRB1. Fulfillment of this criterion shall be considered sufficient evidence that the donor and recipient share one HLA haplotype, and typing of additional family members is not required. * Additional Patient Inclusion Criteria for Patients Assigned to Double Umbilical Cord Blood Arm: 1. Patients must have available two UCB units fulfilling the following criteria: 1. Each unit must have a minimum of 1.5 x 10\^7/kg pre-cryopreserved total nucleated cell dose. For non-red blood cell depleted units, the minimum pre-cryopreserved total nucleated cell dose of each unit must be at least 2.0 x10\^7/kg. 2. Units must be HLA matched at a minimum of 4/6 to the recipient at HLA -A, HLA-B (at low resolution using DNA based typing), and HLA -DRB1 (at high resolution using DNA based typing). 3. Additional graft selection criteria specified in section 2.5 2. Patients must have received at least one cycle of the cytotoxic chemotherapy regimens (or regimen of similar intensity) listed in Appendix D within 3 months of enrollment (measured from the start date of chemotherapy) OR have had an autologous transplant within 24 months of enrollment OR receive 300 cGy as part of the preparative regimen Exclusion Criteria: * Patients with suitably matched related or unrelated donor, as defined per institutional practice. * Recipients of prior autologous hematopoietic stem cell transplantation are ineligible if disease recurrence occurred less than 6 months from their autologous stem cell transplant. * Current uncontrolled bacterial, viral or fungal infection (currently taking medication with evidence of progression of clinical symptoms or radiologic findings). * Prior allogeneic HCT. * Patients with history of primary idiopathic myelofibrosis or any severe marrow fibrosis. * Planned use of prophylactic donor lymphocyte infusion (DLI) therapy. * Anti-donor HLA antibodies. Additional exclusion criteria: * Pregnancy or breast-feeding. * Evidence of HIV infection or known HIV positive serology.