A Myeloablative Conditioning Regimen and Total Body Irradiation Followed by the Transplantation for Patients With Hematological Malignancy

Inclusion Criteria: * Age 4 - 50 years * Patient should not have a related or unrelated volunteer donor that is suitably HLA matched and available in the required time period or be a suitable candidate for an autologous stem cell transplant. * Patients will have one of the following hematological malignancies: Acute myelogenous leukemia (AML): * Complete first remission (CR1) at high risk for relapse as defined by: known prior diagnosis of myelodysplasia (MDS); or therapy related AML; or White cell count at presentation \> 100,000; or Presence of extramedullary leukemia at diagnosis; or Unfavorable FAB type (M0, M5-7); or High-risk cytogenetics (such as those associated with MDS, abnormalities of 5, 7, 8, Philadelphia chromosome, complex karyotype); or High risk molecular markers such as FLT3 mutations; or Requirement for 2 or more inductions to achieve CR1 * Complete second CR (CR2). • Acute lymphoblastic leukemia (ALL): * Complete first remission (CR1) at high risk for relapse as defined by: White cell count at presentation as follows: * \> 100,000 if \< 18 years * \> 50,000 if \> 18 years; or Presence of extensive extra-medullary disease (excluding CNS disease); or Presence of high-risk cytogenetic abnormality such as t(9;22), t(1;19), t(4;11) or other MLL rearrangements (11q23), t(8;14) \[excluding B-ALL in pediatric patients\]; or Failed to achieve complete remission after four weeks of induction therapy Unable to receive required consolidation chemotherapy as would be needed to maintain remission * Complete second or third remission (CR2 or CR3) * Acute undifferentiated leukemia (AUL), infant leukemia, or biphenotypic leukemia in CR1, CR2 or CR3. Patients with infant leukemia must be eligible to receive total body irradiation. * Juvenile Myelomonocytic leukemia (JMML) with \< 30% bone marrow blasts. * Chronic myelogenous leukemia (CML) * GleevecTM failure in 1st or 2nd chronic phase; * Gleevec failure in 1st or 2nd accelerated phase. • Myelodysplastic Syndrome (MDS) with one of the following: * Low (Score 0) International Prognostic Scoring System (IPSS) score with Life-threatening cytopenia(s); or Red cell or platelet transfusion dependent * Intermediate (Score 1) or High (Score 2) International Prognostic Scoring System (IPSS) score. * Patients with bone marrow blasts \> 10% should have AML induction therapy with disease response to \< 5% blasts and at least partial count recovery. • Non-Hodgkin's Lymphoma * Patients with high-grade disease following initial therapy and are not appropriate for further chemotherapy or autologous stem cell transplantation. * Patients with high-grade or diffuse large cell NHL with recurrent disease after first remission and must have: Chemo-sensitivity as evidenced by \> partial remission (PR) (defined as \> 50% reduction in mass size after therapy). * Patients with adequate organ function and performance status criteria as measured by: * Karnofsky score \> or = to 70 % or Lansky score \> or = to 70% * Renal: Calculated creatinine clearance \> or = to 60 ml/min * Hepatic: Total bilirubin \< 2.5 mg/dL unless benign congenital hyperbilirubinemia (Gilbert's syndrome) and ALT/AST \< 3 x upper limit of normal * Albumin \> or = to 2.5 * Pulmonary: Pulmonary function (spirometry and corrected DLCO) \> or = to 60% normal if available (in small children use History and Physical and CT scan as necessary to determine pulmonary status) * Cardiac: Left ventricular ejection fraction \> or = to 50% * Double Unit Umbilical Cord Blood Grafts: Units will be selected based on the HLA match to the patient and on the basis of the individual and combined cell doses of the units. * Patient has a related or unrelated volunteer donor that is suitably HLA matched and available in the required time period. * Patient is a candidate for an autologous stem cell transplant. * Active CNS leukemia. * Acute Myelogenous Leukemia in greater than CR2. * Acute Myelogenous Leukemia evolved from myelofibrosis. * Acute Lymphoblastic Leukemia, acute undifferentiated leukemia, biphenotypic leukemia or infant leukemia greater than CR3. * Any acute leukemia with: * Morphologic relapse or persistent disease in the BM (cytogenetic relapse without morphologic evidence of relapse or cytogenetic persistent disease in the BM is acceptable); or * Active extra-medullary leukemia; or * Requiring greater than 2 cycles of chemotherapy to obtain present remission status; * Bone Marrow aplasia (defined as BM cellularity less than 5% at transplant work-up); or * MDS with greater than 10% bone marrow blasts refractory to chemotherapy; or * CML in blast crisis; or * NHL refractory to chemotherapy (less than PR after 2 or more regimens); or * Prior autologous or allogeneic HSC transplant at any time; or * Prior radiation therapy rendering patient ineligible for TBI; or * Uncontrolled viral, bacteria or fungal infection at time of study enrollment; or * Seropositive or NAT positive for HIV; or * Females who are pregnant or breast feeding; or * Patient or guardian unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, follow-up, and research tests.