Treg Cells for AGVHD in Non-myeloablative UCB Transplant

Inclusion Criteria: * Must be ≥18, but \< 70 years of age with no matched 5/6 or 6/6 sibling donor - patients ≥ 70 and ≤ 75 years of age may be eligible if they have a Co-Morbidity score ≤ 2 (Appendix II) * Three UCB units composing the graft will be selected according to the current University of Minnesota umbilical cord blood graft selection algorithm. * Each UCB unit must be matched at 4-6 HLA-A, B, DRB1 antigens with the recipient. This may include 0-2 antigen mismatches at the A or B or DRB1 loci. Each unit must be a 4-6 HLA-A, B, DRB1 antigen match to each other, not necessarily at the same loci they are matched to the recipient. * Disease Criteria * Acute Leukemias: Must be in remission by morphology (\<5% blasts). Note cytogenetic relapse or persistent disease without morphologic relapse is acceptable. Also a small percentage of blasts that is equivocal between marrow regeneration versus early relapse are acceptable provided there are no associated cytogenetic markers consistent with relapse. * 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 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 Myelogenous Leukemia (AML) in first complete remission (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 in first or subsequent CR, adult T-cell leukemia/lymphoma in first or subsequent complete remission (CR) * Burkitt's Lymphoma in CR2 or subsequent complete remission (CR) * Natural Killer Cell Malignancies * Chronic Myelogenous Leukemia: all types except refractory blast crisis. Chronic phase patients must have failed or been intolerant to at least one tyrosine-kinase inhibitor * Myelodysplastic Syndrome: any subtype including refractory anemia (RA) if severe pancytopenia or complex cytogenetics. Blasts must be less than 5%. If 5% or more requires induction therapy pre-transplant to reduce blast count to ≤5%. * Large-Cell Lymphoma, Hodgkin Lymphoma and Multiple Myeloma with chemotherapy sensitive disease that has failed or patients who are ineligible for an autologous transplant. * Chronic Lymphocytic Leukemia/Small Lymphocytic Lymphoma (CLL/SLL), Marginal Zone B-Cell Lymphoma, Follicular Lymphoma, which have progressed within 12 months of achieving a partial or complete remission. Patients who had remissions lasting \> 12 months, are eligible after at least two prior therapies. Patients with bulky disease should be considered for debulking chemotherapy before transplant. Patients with refractory disease are eligible, unless has bulky disease and an estimated tumor doubling time of less than one month. * Lymphoplasmacytic Lymphoma, Mantle-Cell Lymphoma, Prolymphocytic Leukemia are eligible after initial therapy if chemotherapy sensitive. Patients must have undergone an autologous transplant ≤ 12 months prior to allogeneic transplantation or have received multi-agent or immunosuppressive chemotherapy within 3 months of the preparative regimen. * Adequate performance status defined as a Karnofsky score ≥ 70% * Adequate organ function within 14 days (30 days for cardiac and pulmonary) of enrollment defined as: * Renal: creatinine \< 2.0 mg/dL, for patient with a creatinine \> 1.2 mg/dL or a history of renal dysfunction an estimated glomerular filtration rate \> 40 mL/min/1.73 m2 is required * Hepatic: bilirubin, AST, ALT, alkaline phosphatase \< 5 x upper limit of normal, * Pulmonary function: DLCOcorr \> 40% normal, * Cardiac: left ventricular ejection fraction \> 35% * Voluntary written consent signed before performance of any study-related procedure not part of normal medical care Exclusion Criteria: * Untreated active infection at time of transplantation * History of HIV infection * Pregnant or breast feeding. The agents used in this study may be teratogenic to a fetus and there is no information on the excretion of agents into breast milk. Females of childbearing potential must have a blood test or urine study within 14 days prior to registration to rule out pregnancy * Prior allogeneic transplantation * Less than 3 months from myeloablative conditioning for autologous transplantation