Inclusion Criteria:

          -  Patient must have hematologic malignancy that meets institutional eligibility
             requirements for cord blood transplant

          -  Malignancies included are:

               -  Acute leukemia, including Acute myeloid leukemia (AML), biphenotypic acute
                  leukemia or mixed-lineage leukemia, acute lymphoblastic leukemia (ALL); all
                  patients must be in complete response (CR) as defined by < 5% blasts by
                  morphology/flow cytometry in a representative bone marrow sample with adequate
                  cellularity to assess remission status

               -  Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate
                  (Int)-2 or high risk (i.e., RAEB, RAEBt) or refractory anemia with severe
                  pancytopenia or high risk cytogenetics; blasts must be < 10% in a representative
                  bone marrow aspirate

               -  Chronic Myeloid Leukemia excluding refractory blast crisis; to be eligible in
                  first chronic phase (CP1) patient must have failed or be intolerant to tyrosine
                  kinase inhibitor therapy

          -  High dose TBI regimen: 18 to =< 45 years

          -  Intermediate intensity regimen: 18 =< 65 years

          -  Patients 18 to =< 45 years: Karnofsky (>= 18 years old) >= 70 or Eastern Cooperative
             Oncology Group (ECOG) 0-1

          -  Patients > 45 to =< 65 years: Karnofsky >= 70 or ECOG 0-1 and non-age adjusted
             comorbidity index =< 5

          -  Calculated creatinine clearance must be > 60 mL and serum creatinine =< 2 mg/dL

          -  Total serum bilirubin must be < 3 mg/dL unless the elevation is thought to be due to
             Gilbert's disease or hemolysis

          -  Transaminases must be < 3 x the upper limit of normal per reference values of treating
             institution

          -  Carbon monoxide diffusing capability (DLCO) corrected >= 60% normal (may not be on
             supplemental oxygen)

          -  Left ventricular ejection fraction >= 50% OR

          -  Shortening fraction > 26%

          -  Ability to understand and the willingness to sign a written informed consent form

          -  DONOR: Minimum requirement: The cord blood (CB) unit must be matched at a minimum at
             4/6 HLA-A, B antigens and DRB1 allele with the recipient; therefore, 0-2 mismatches at
             the A or B or DRB1 loci based on intermediate resolution at HLA-A, B and high
             resolution allele level typing at HLA- DRB1 are allowed

          -  DONOR: Institutional guidelines for HLA-match may be followed as long as the minimum
             criteria for HLA-matching as above are met

          -  DONOR: The CB unit selected for transplant must have a MINIMUM of 2.5 x 10^7 TNC/kg

          -  DONOR: The minimum recommended CD34/kg cell dose is 1.7 x 10^5 CD34/kg

          -  DONOR: A domestic backup unit, meeting the same HLA and cell dose requirements, must
             be identified and reserved prior to the start of the treatment plan for possible
             infusion in the unlikely event of poor post-thaw viability of the primary CB unit

        Exclusion Criteria:

          -  Uncontrolled viral or bacterial infection at the time of study enrollment

          -  Active or recent (prior 6 month) invasive fungal infection unless cleared by
             innovation and development (ID) consult

          -  History of human immunodeficiency virus (HIV) infection

          -  Pregnant or breastfeeding

          -  Prior allogeneic transplant

          -  Central nervous system (CNS) leukemic involvement not clearing with intrathecal
             chemotherapy; diagnostic lumbar puncture is to be performed