UCB Transplant for Hematological Diseases Using a Non Myeloablative Prep

Inclusion Criteria: * Age, Performance Status, and Graft Criteria * \<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 (http://www.qxmd.com/calculate-online/hematology/hct-ci) * Karnofsky score ≥ 70% (≥ 16 years) or Lansky score ≥ 50 (\< 16 years) * UCB graft selected according to current University of Minnesota umbilical cord blood graft selection algorithm * Eligible Diseases All diseases listed below are advanced hematologic malignancies not curable by conventional chemotherapy. Responses to conventional treatment range from zero to 30% but are typically short lived. * 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 vs. early relapse are acceptable provided there are no associated cytogenetic markers consistent with relapse. * Acute Myeloid Leukemia (AML) and related precursor neoplasms: 2nd or greater complete remission (CR); first complete remission (CR1) in patients \> 60 years old; CR1 in ≤ 60 years old that is NOT considered as favorable-risk. Favorable risk AML is defined as having one of the following: * t(8,21) without cKIT mutation * inv(16) or t(16;16) without cKIT mutation * Normal karyotype with mutated NPM1 and wild type FLT-ITD * Normal karyotype with double mutated CEBPA * Acute prolymphocytic leukemia (APL) in first molecular remission at the end of consolidation * Acute lymphoblastic leukemia (ALL)/lymphoma: second or greater CR; CR1 unable to tolerate consolidation chemotherapy due to chemotherapy-related toxicities; CR1 high-risk ALL. High risk ALL is defined as having one of the following: * Evidence of high risk cytogenetics, e.g. t(9;22), t(1;19), t(4;11), other MLL rearrangements, IKZF1 * 30 years of age or older at diagnosis * White blood cell counts of greater than 30,000/mcL (B-ALL) or greater than 100,000/mcL (T-ALL) at diagnosis * CNS leukemia involvement during the course of disease * Slow cytologic response (\>10% lymphoblasts in bone marrow on Day 14 of induction therapy) * Evidence of persistent immonophenotypic or molecular minimal residual disease (MRD) at the end of induction and consolidation therapy * Biphenotypic/Undifferentiated/Prolymphocytic Leukemias in first or subsequent CR * Chronic myelogenous leukemia in chronic or accelerated phase, or CML blast crisis in morphological remission (\<5% blasts): Chronic phase patients must have failed at least two tyrosine kinase inhibitors, been intolerant to all available TKIs, or have T315I mutation. * Myelodysplastic syndrome: IPSS INT-2 or High Risk; R-IPSS High or Very High; WHO classification: RAEB-1, RAEB-2; Severe Cytopenias: ANC \< 0.8, Anemia or thrombocytopenia requiring transfusion; Poor or very poor risk cytogenetics based on IPSS or R-IPSS definitions; therapy-related MDS. Blasts must be \< 5% by bone marrow aspirate morphology. If ≥5% blasts, patient requires chemotherapy for cytoreduction to \<5% blasts prior to transplantation. * MRD positive leukemia (AML, ALL or accelerated/blast phase CML). Selected patients in morphologic CR, but with positive immunophenotypic (flow cytometry) or molecular evidence of MRD may be eligible if recent chemotherapy has not resulted in MRD negative status. * Leukemia or MDS in aplasia. These patients may be taken to transplant if after induction therapy they remain with aplastic bone marrow and no morphological or flow-cytometry evidence of disease ≥ 28 days post-therapy. These high risk patients will be analyzed separately. * Burkitt's lymphoma in CR2 or subsequent CR * Relapsed T-Cell Lymphoma that is chemotherapy sensitive in CR/PR that has failed or ineligible for an autologous transplant. * Natural killer cell malignancies * Large-cell lymphoma, Hodgkin lymphoma and multiple myeloma with chemotherapy sensitive disease who are ineligible for an autologous transplant. * Relapsed 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 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. * Relapsed Multiple Myeloma that is chemotherapy sensitive and has failed or ineligible for an autologous transplant. * Plasma Cell Leukemia after initial therapy if achieved at least in partial remission; or relapsed and achieved subsequent remission (CR/PR) * Acquired Bone marrow failure syndromes, except for Fanconi anemia * Myeloproliferative Neoplasms/Myelofibrosis * Other Leukemia Subtypes: A major effort in the field of hematology is to identify patients who are of high risk for treatment failure so that patients can be appropriately stratified to either more (or less) intensive therapy. This effort is continually ongoing and retrospective studies identify new disease features or characteristics that are associated with treatment outcomes. Therefore, if new features are identified after the writing of this protocol, patients can be enrolled with the approval of two members of the study committee. * Additional Criteria for Bulky Disease (lymphomas) * If stable disease is best response, the largest residual nodal mass must \< 5 cm (approximately) * If response to previous therapy, the largest residual mass must represent a 50% reduction and be \< 7.5 cm (approximately) * Organ Function Criteria Adequate organ function is defined as: * Cardiac: Absence of decompensated congestive heart failure, or uncontrolled arrhythmia and left ventricular ejection fraction \> 40%. For children that are not able to cooperate with MUGA and echocardiography, such should be clearly stated in the physician's note. * Pulmonary: DLCO, FEV1, FVC ≥ 40% predicted, and absence of O2 requirements. For children that are not able to cooperate with PFTs, a pulse oximetry with exercise should be attempted. If neither test can be obtained it should be clearly stated in the physician's note. * Liver: Transaminases ≤ 5 x upper limit of normal (ULN) and total bilirubin ≤ 2.5 mg/dL except for patients with Gilbert's syndrome or hemolysis * Renal: Creatinine ≤ 2.0 mg/dl (adults) and creatinine clearance ≥ 40 mL/min (pediatrics). Adults with a creatinine \> 1.2 mg/dl or a history of renal dysfunction must have estimated creatinine clearance ≥ 40 ml/min/1.73m\^2. Adequate performance status is defined as Karnofsky score ≥ 70% (≥ 16 years of age) or Lansky score ≥ 50 (pediatrics) * Sexually active females of childbearing potential and males with partners of child-bearing potential must agree to use adequate birth control during study treatment. * Voluntary written consent (adult or parent/guardian with presentation of the minor information sheet, if appropriate) Exclusion Criteria: * Pregnant or breast feeding. The agents used in this study include Pregnancy Category D: known to cause harm to a fetus. Females of childbearing potential must have a negative pregnancy test prior to starting therapy. * Untreated active infection * Active HIV infection or known HIV positive serology * Less than 3 months since prior myeloablative transplant * Evidence of progressive disease by imaging modalities or biopsy - persistent PET activity, though possibly related to lymphoma, is not an exclusion criterion in the absence of CT changes indicating progression. * CML in blast crisis * Large cell lymphoma, mantle cell lymphoma and Hodgkin disease that is progressing on salvage therapy. * Active central nervous system malignancy