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). Low resolution using DNA based typing at HLA-A, -B, and -DRB1 for potential haploidentical 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. - Patients must have received either: a)At least one cycle of a the following cytotoxic chemotherapy regimen (or regimen of similar intensity) within 3 months of enrollment (measured from the start date of chemotherapy)(1. multiagent chemotherapy, 2. chemotherapy regimens like those that are given as induction or consolidation of acute leukemia, 3. single drug alkylator agent. 4. single agent alemtuzumab or brentuximab vedotin); or, b) Autologous HCT greater than 6 months and less than 2 years prior to enrollment. - 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 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 - 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-IND, normal karyotype with double mutated CEBPA, APL in first molecular remission at end of consolidation - Acute Leukemias in 2nd or subsequent CR - Biphenotypic/Undifferentiated/Prolymphocyctic 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: a) Chemotherapy-sensitive (complete or partial response; lymphomas that have failed at least 1 prior regimen of multi-agent chemotherapy and are INELIGIBLE for an autologous transplant; b) Marginal zone B-cell lymphoma or follicular lymphoma that has progressed after at least at least two prior therapies (excluding single agent Rituxan). 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. ALT, AST, and Alkaline Phosphatase less than 5 x ULN; 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: DLCO (corrected for hemoglobin), FEV1, and FVC greater than 50% predicted; e. Performance status: Karnofsky score greater than or equal to 70%. - 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: Patients must have available two UCB units fulfilling the following criteria: a. 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; and, b. 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. Exclusion Criteria: - Patients with suitably matched related or unrelated donor. - Autologous hematopoietic stem cell transplant less than 6 months prior to enrollment. - Pregnancy or breast-feeding. - Evidence of HIV infection or known HIV positive serology. - 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.