Treatment of Single or Double Umbilical Cord Trans + Graft-versus-host Disease (GVHD) Prophylaxis w/ Tacrolimus & Mycophenolate Mofetil

Patient and UCB Unit Selection: Inclusion Criteria: General (Adults and Pediatrics) Only one of the following should be present: * Acute leukemia (lymphocytic or myeloid or undifferentiated or biphenotypic) in complete remission 2 or beyond * Acute lymphocytic leukemia, Philadelphia chromosome positive in complete remission 1 or beyond * Acute myeloid leukemia in complete remission 1 if it has evolved from a myeloproliferative disorder (MPD) or myelodysplastic syndrome (MDS). * Acute leukemia in complete remission 1 if there is a failure to recover normal blood counts or the development of MDS following induction chemotherapy. * Therapy related acute leukemia in complete remission 1 or beyond * Chronic myeloid leukemia (CML) chronic phase-1 (imatinib failures, imatinib intolerance), or any CML beyond first chronic phase * Myelodysplastic syndromes (Intermediate -1 or higher risk by IPSS) * Therapy related MDS (irrespective of IPSS) * Multiple myeloma must have had prior chemotherapy or autologous transplant * Chronic lymphocytic leukemia must have failed two lines of conventional therapy but still chemosensitive to third line therapy. * Chemosensitive Non-Hodgkin's lymphoma or Hodgkin's lymphoma in CR or PR after failing induction therapy. * High risk acute leukemia/lymphoma eg Nk/T cell, HTLV associated leukemia/lymphoma, other T cell lymphoma/leukemia in first best response * For patients with acute leukemia-they must be in a remission (less than 5% leukemic marrow blasts) at time of study entry. Inclusion Criteria (Adults - 18 years or older) * Karnofsky score of \> 70% * Estimated creatinine clearance of \> 60 ml/min * Left ventricular ejection fraction of \>50% * Pulmonary function test with DLCO, FEV1 and FVC of \>60% * Total bilirubin and SGOT of \< 3.0 x upper limits of normal * Note: Age 18- 40 years for adult myeloablative conditioning Age \> 40 -50 years for adult reduced intensity conditioning Inclusion Criteria (Pediatrics - 18 years and younger) * Karnofsky or Lansky score of \> 70% * Estimated Creatinine clearance of \> 60 ml/min * Left ventricular ejection fraction of \>50% * Pulmonary function test with FEV1 and FVC of \>60% (for patients \>6 years of age) * Total bilirubin and SGOT of \< 3.0 x upper limits of normal * Note: All pediatric patients will receive myeloablative conditioning Inclusion Criteria - Donor Issues * No available HLA identical or 1 antigen/allele mismatched (Class I-A, B or Class II DR locus) related donor Inclusion Criteria: Umbilical Cord Blood Unit-HLA Typing * At least a HLA 4/6 match (Class I-A, B by low resolution, Class II-DR by high resolution) to recipient * For double UCB SCT each unit should be at least a 4/6 match (Class I-A,B by low resolution, Class II-DR by high resolution) to recipient, and should be at least a 4/6 match (Class I-A,B by low resolution, Class II-DR by high resolution) to each other Inclusion Criteria: Umbilical Cord Blood Unit-Cell dose * For Single UCB SCT: the unit will have ≥ 3.5 X 107 NC/kg of recipient body weight (For pediatric patients a cell dose ≥ 3.0 X 107 NC/kg of recipient body weight is acceptable). Recipient body weight will be determined as per standard guidelines. * For Double UCB SCT: (done only if no single UCB unit ≥ 3.5 X 107 NC/kg of recipient body weight is available for adults, and ≥ 3.0 X 107 NC/kg of recipient body weight is available for pediatric patients ) * The larger of the two units (UCB1) will have a minimum cell dose of 2.0 X 107 NC/kg of recipient body weight. The smaller of the two units (UCB2) will have a minimum of 0.5 X 107 NC/kg of recipient body weight. The total cell dose UCB1 + UCB2 will be ≥ 2.5 X 107 NC/kg of recipient body weight. -Adult patients eligible for a double UCB SCT but without an appropriate second UCB unit will be enrolled in the study if their single UCB unit contains ≥ 2.5 x 107 NC/kg recipient body weight. Exclusion Criteria * Organ dysfunction as per standard guidelines. Unable to give informed consent (for adults only) * Pregnant or lactating * Sexually active individuals capable of becoming pregnant or causing a pregnancy who are unable or unwilling to use appropriate contraceptives. * Active use of illicit drugs as evidenced by a positive toxicology screen for a substance not prescribed by a medical professional just prior to initiating the preparative regimen * Actively smoking as evidenced by a positive nicotine screen just prior to initiating the preparative regimen * HIV positive * Patients with other unrelated malignancies will be excluded except: * diagnosis of skin cancer (squamous cell or basal cell) * diagnosis of cervical dysplasia (CIN I-III) * any other malignancy which is currently in remission and was treated with curative intent more than 5 years preceding study entry * In patients with secondary MDS or secondary acute leukemias-the previous non-hematopoietic neoplasm should be in remission but can be within 5 years of study entry