Inclusion Criteria:

- Graft Criteria

- The unrelated cord blood donor(s) must be 4-6/6 HLA-A, B, DRB1 matched with the
recipient (HLA matching using molecular techniques: A and B to antigen level
resolution and DR to allele level resolution).

- No existing HLA-identical related donor is available.

- Suitable UCB units available according to Umbilical Cord Blood Graft (UCB)
selection algorithm. The UCB graft may consist of one or two UCB units.

- Age and Disease Criteria

- Patients aged ≤ 55 years must have a hematological malignancy as described
below:

- Acute myeloid leukemia (AML): high risk CR1 (as evidenced by preceding
myelodysplastic syndrome [MDS], high risk cytogenetics, ≥ 2 cycles to obtain
complete remission [CR], erythroblastic or megakaryocytic leukemia; CR2+. All
patients must be in CR as defined by hematological recovery, AND <5% blasts by
light microscopy within the bone marrow with a cellularity of ≥15%.

- Very high risk pediatric patients with AML. Patients <21 years, however, are
eligible with (M2 marrow) with < or = 25% blasts in marrow after having failed
one or more cycles of chemotherapy. This group of patients will be analyzed
separately.

- Acute lymphocytic leukemia (ALL): high risk CR1 [t(9;22), t (1:19), t(4;11) or
other MLL rearrangements] hypodiploidy, or IKZF1 abnormalities), DNA index <
0.81, > 1 cycle to obtain CR or presence minimal residual disease (MRD).
Patients in CR2+ are eligible. All patients must be in CR as defined by
hematological recovery, AND <5% blasts by light microscopy within the bone
marrow with a cellularity of ≥15%.

- Very high risk pediatric patients with ALL. patients <21 years are also
considered high risk CR1 if they had M2 or M3 marrow at day 42 from the
initiation of induction or M3 marrow at the end of induction. They are eligible
once they achieved a complete remission

- 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.

- Plasma Cell leukemia after initial therapy, who achieved at least a partial
remission

- Advanced myelofibrosis

- Myelodysplasia (MDS) IPSS Int-2 or High risk (i.e. RAEB, RAEBt) or refractory
anemia with severe pancytopenia or high risk cytogenetics. Blasts must be < 10%
by a representative bone marrow aspirate morphology.

- Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL), marginal zone
B-cell lymphoma or follicular lymphoma are eligible if there was disease
progression/relapse within 12 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 (nodal mass greater than 5 cm)
should be considered for debulking chemotherapy before transplant.

- Lymphoplasmacytic lymphoma, mantle-cell lymphoma, prolymphocytic leukemia are
eligible after initial therapy in CR1+ or PR1+.

- Large cell NHL > CR2/> PR2. Patients in CR2/PR2 with initial short remission (<6
months) are eligible.

- Lymphoblastic lymphoma, Burkitt's lymphoma, and other high-grade NHL after
initial therapy if stage III/IV in CR1/PR1 or after progression if stage I/II <
1 year.

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

- Organ function and Performance Status Criteria

- Recipients must have a Karnofsky score (adults) ≥ 80 % or Lansky score ≥ 50%
(pediatrics) and have acceptable organ function defined as:

- Renal: creatinine < or = 2.0 (adults) or creatinine clearance > 40 ml/min
(pediatrics)

- Hepatic: bilirubin, aspartate aminotransferase/alanine aminotransferase
(AST/ALT), Alkaline phosphatase (ALP) < or = 5 x upper limit of normal

- Pulmonary function: Diffusion capacity corrected (DLCOcorr) > 50% normal

- Cardiac: left ventricular ejection fraction > or = 45%.

- Voluntary written informed consent before performance of any study-related
procedure not part of normal medical care.

Exclusion Criteria

- Active infection at time of transplantation (including active infection with
Aspergillus or other mold within 30 days.)

- History of human immunodeficiency virus (HIV) infection

- Pregnant or breast feeding. The agents used in this study may be teratogenic to a
fetus and there is no information on the excretion of agents into breast milk All
females of childbearing potential must have a blood test or urine study within 2
weeks prior to registration to rule out pregnancy.

- Chemotherapy refractory large cell and high grade NHL (ie progressive disease after >
2 salvage regimens)

- If < or = 18 years old, prior myeloablative transplant within the last 6 months. If
>18 years old prior myeloablative allotransplant or autologous transplant

- 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 tostumomab (Bexxar),
as part of their salvage therapy are not eligible for myeloablative umbilical cord
blood transplant.