Inclusion Criteria:

- GRAFT CRITERIA:

- UCB units will be selected according to current umbilical cord blood graft
selection algorithm; one or 2 UCB units may be used to achieve the required cell
dose

- The UCB graft is matched at 4-6 human leukocyte antigen (HLA)-A, B, DRB1
antigens with the recipient; this may include 0-2 antigen mismatches at the A or
B or DRB1 loci; unit selection based on cryopreserved nucleated cell dose and
HLA-A,B, DRB1 using intermediate resolution A, B antigen and DRB1 allele typing

- If 2 UCB units are required to reach the target cell dose, each unit must be a
4-6 antigen match to the recipient

- Acute myeloid leukemia (AML):

- High risk first complete remission (CR1) as evidenced by preceding
myelodysplastic syndromes (MDS), high risk cytogenetics (for example, monosomy 5
or 7), or high risk as defined by referring institution treatment protocol, >= 2
cycles to obtain complete response (CR), erythroblastic or megakaryocytic
leukemia; >= second complete remission (CR2)

- All patients must be in CR as defined by hematologic recovery and < 5% blasts by
morphology within the bone marrow and a cellularity of >= 15%

- Patients in which adequate marrow/biopsy specimens cannot be obtained to
determine remission status by morphologic assessment, but have fulfilled
criteria of remission by flow cytometry, recovery of peripheral blood counts
with no circulating blasts, and/or normal cytogenetics (if applicable) may still
be eligible; reasonable attempts must be made to obtain an adequate specimen for
morphologic assessment, including possible repeat procedures

- Very high risk pediatric/young adult patients with AML: Patients < 25 years, however,
are eligible with (M2 marrow) with =< 25% blasts in marrow after having failed one or
more cycles of chemotherapy; this group of patients will be analyzed separately

- Acute Lymphoblastic Leukemia (ALL):

- High risk CR1 [for example, but not limited to: t(9;22), t(1;19), t(4;11) or
other mixed lineage leukemia (MLL) rearrangements, hypodiploid]

- Greater than 1 cycle to obtain CR

- >= CR2

- All patients must be in CR as defined by hematologic recovery and < 5% blasts by
morphology within the bone marrow and a cellularity of >= 15%

- Patients in which adequate marrow/biopsy specimens cannot be obtained to
determine remission status by morphologic assessment, but have fulfilled
criteria of remission by flow cytometry, recovery of peripheral blood counts
with no circulating blasts, and/or normal cytogenetics (if applicable) may still
be eligible; reasonable attempts must be made to obtain an adequate specimen for
morphologic assessment, including possible repeat procedures

- Chronic myelogenous leukemia (CML) excluding refractory blast crisis; to be eligible
in first chronic phase (CP1) patient must have failed or be intolerant to imatinib
mesylate

- Advanced myelofibrosis

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

- Lymphoblastic lymphoma, Burkitt's lymphoma, and other high-grade NHL after initial
therapy if stage III/IV in first partial response (PR1) or after progression if stage
I/II < 1 year; stage III/IV patients are eligible after progression in CR/PR

- Chronic lymphocytic leukemia /small lymphocytic lymphoma (CLL/SLL), marginal zone
B-cell lymphoma, lymphoplasmacytic lymphoma or follicular lymphoma that have
progressed after at least two different prior therapies; patients with bulky disease
(nodal mass greater than 5 cm) should be considered for debulking chemotherapy before
transplant; these patients must be presented at primary care center (PCC) prior to
enrollment, given potential competing eligibility on autotransplant protocols

- Mantle-cell lymphoma, prolymphocytic leukemia: Eligible after initial therapy in >=
CR1 or >= PR1

- Large cell Non-Hodgkin lymphoma (NHL):

- Patients in CR2/second partial response (PR2) with initial short remission (< 6
months) are eligible

- These patients must be presented at PCC prior to enrollment, given potential
competing eligibility on autotransplant protocols

- Multiple myeloma beyond PR2: Patients with chromosome 13 abnormalities, first
response lasting less than 6 months, or beta-2 microglobulin > 3 mg/L, may be
considered for this protocol after initial therapy

- Performance status score: Karnofsky (for adults) >= 70% or Lansky (for children) >=
50%

- Creatinine < 2.0 mg/dL (for adults) or creatinine clearance > 60 ml/min (for
children)

- Patients with clinical or laboratory evidence of liver disease will be evaluated for
the cause of liver disease, its clinical severity in terms of liver function,
histology, and the degree of portal hypertension; patients with fulminant liver
failure, cirrhosis with evidence of portal hypertension or bridging fibrosis,
alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices,
hepatic encephalopathy, or correctable hepatic synthetic dysfunction evidenced by
prolongation of the prothrombin time, ascites related to portal hypertension,
bacterial or fungal abscess, biliary obstruction, chronic viral hepatitis with total
serum bilirubin > 3mg/dL, and symptomatic biliary disease will be excluded

- Diffusion capacity for carbon monoxide corrected (DLCOcorr) > 50% normal

- Left ventricular ejection fraction > 45% or shortening fraction > 26%

Exclusion Criteria:

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

- Active or recent (prior 6 month) invasive fungal infection without interdisciplinary
(ID) consult and approval

- History of human immunodeficiency virus (HIV) infection

- Pregnant or breastfeeding

- Chemotherapy refractory large cell and high grade NHL (i.e., progressive disease
after > 2 salvage regimens)

- Prior myeloablative transplant containing full dose TBI (greater than 8 Gray [Gy])

- Any prior myeloablative transplant within the last 6 months

- Extensive prior therapy including > 12 months alkylator therapy or > 6 months
alkylator therapy with extensive radiation

- Patients who have received Y-90 ibritumomab (Zevalin) or I-131 tositumomab (Bexxar),
as part of their salvage therapy are not eligible for myeloablative umbilical cord
blood transplant