Inclusion Criteria:

Age, Graft Cell Dose and Graft HLA Criteria (all patients: Arms 1, 2, 3)

- Subjects must be <70 years old. Subjects ages ≥ 70 and ≤ 75 may be eligible if they
have a Co-Morbidity Scoring (HCT-CI) score ≤ 2.

- Umbilical cord blood (UCB) units will be selected according to current University of
Minnesota 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 HLA-A, B, DRB1 antigens with the recipient. This may
include 0-2 antigen mismatches at the A or B or DRB1 loci. If 2 UCB units are
required to reach the target cell dose, each unit must be a 4-6 HLA-A, B, DRB1
antigen match to each other, as well as a 4-6 antigen match to the recipient.

- Patients co-enrolled in MT-2006-01 Phase I Study of Infusion of Umbilical Cord Blood
Derived CD25+CD4+ T-Regulatory (Treg) Cells after Non-Myeloablative Cord\Blood
Transplantation will receive grafts composed of 2 UCB units.

Disease Criteria: 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: high risk complete remission 1 (CR1) (as evidenced by
preceding myelodysplastic syndrome (MDS), high risk cytogenetics such as those
associated with MDS or complex karyotype, > 2 cycles to obtain CR or
erythroblastic and megakaryocytic); second or greater CR.

- Acute lymphoblastic leukemia/lymphoma: high risk CR1 as evidenced by high risk
cytogenetics (e.g. t(9;22), t(1;19),t(4;11), other myeloid/lymphoid or mixed
lineage leukemia [MLL] rearrangements, hypodiploidy or Ikaros family zinc finger
1 [IKZF1]), > 1 cycle to obtain CR or evidence of minimal residual disease
(MRD). Patients in second or greater CR are also eligible.

- Burkitt's lymphoma in CR2 or subsequent CR

- Natural Killer cell malignancies

- Chronic myelogenous leukemia: all types except refractory blast crisis. Chronic phase
patients must have failed or been intolerant to Gleevec

- Myelodysplastic syndrome: any subtype including refractory anemia (RA) if severe
pancytopenia or complex cytogenetics. Blasts must be less than 5%. If 5% or more
requires induction therapy pre-transplant to reduce blast count to

≤5%.

- Large-cell lymphoma, Hodgkin lymphoma and multiple myeloma with chemotherapy
sensitive disease that has failed or patients who are ineligible for an autologous
transplant.

- 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 has 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.

- Refractory leukemia or MDS. These patients may be taken to transplant in aplasia
after induction or re-induction chemotherapy or radiolabeled antibody. These high
risk patients will be analyzed separately (Arm 3).

- Bone marrow failure syndromes, except for Fanconi Anemia

- Myeloproliferative syndromes Patients who have undergone an autologous transplant >12
months prior to allogeneic transplantation and who have not received multi-agent or
immunosuppressive chemotherapy within the preceding 3 months must receive ATG as part
of the preparative regimen (Arm 2).

Organ Function and Performance Status Criteria (all patients)

- Adequate organ function is defined as:

- Cardiac: Absence of decompensated congestive heart failure, or uncontrolled
arrhythmia and left ventricular ejection fraction > 35%. For children that are
not able to cooperate with multigated acquisition scan (MUGA) and
echocardiography, such should be clearly stated in the physician's note

- Pulmonary: Carbon Monoxide Diffusing Capacity (DLCO) > 30% predicted, and
absence of oxygen (O2) requirements. For children that are not able to cooperate
with pulmonary function tests (PFTs), a pulse oximetry with exercise should be
attempted. If nether test can be obtained it should be clearly stated in the
physician's note.

- Liver: Transaminases < 5 x upper limit of normal and bilirubin < 3 x upper limit
of normal

- Renal: Creatinine < 2.0 mg/dl (adults) and creatinine clearance > 40 ml/min
(pediatrics). Adults with a creatinine > 1.2 or a history of renal dysfunction
must have estimated creatinine clearance > 40 ml/min.

- Adequate performance status is defined as Karnofsky score > or = 60% or Lansky
score > or = 50 (pediatrics)

Other Inclusion Criteria (all patients)

- If recent mold infection (e.g. Aspergillus) - must have minimum of 30 days of
appropriate treatment before bone marrow transplant (BMT) and infection controlled
and be cleared by Infectious Disease.

- Second BMT: Must be > 3 months after prior myeloablative transplant.

- Patients must be ineligible for autologous transplantation due to prior autologous
transplant, an inadequate autologous stem cell harvest, inability to withstand a
myeloablative preparative regimen, or clinically aggressive/high risk disease.

- Patients are eligible for transplantation if there is no evidence of progressive
disease by imaging modalities or biopsy. Persistent positron emission tomography scan
(PET) activity, though possibly related to lymphoma, is not an exclusion criterion in
the absence of computated tomography scan (CT) changes indicating progression.

- Patients with stable disease are eligible for transplantation if the largest residual
nodal mass is < 5 cm (approximately). For patients who have responded to preceding
therapy, the largest residual mass must represent a 50% reduction and be < 7.5 cm
(approximately).

Exclusion Criteria:

- < 70 years with an available 5-6/6 HLA-A, B, DRB1 matched sibling donor

- Pregnancy or breastfeeding

- Evidence of human immunodeficiency virus (HIV) infection or known HIV positive
serology

- Current active serious infection

- Unless in post-chemotherapy and radioimmunoconjugated antibody induced aplasia, when
he/she would be eligible for Arm 3, patients with acute leukemia in morphologic
relapse/ persistent disease defined as > 5% blasts in normocellular bone marrow OR
any % blasts if blasts have unique morphologic markers (e.g. Auer rods) or associated
cytogenetic markers that allows morphologic relapse to be distinguished are not
eligible.

- Chronic myelogenous leukemia (CML) in refractory blast crisis

- Large cell lymphoma, mantle cell lymphoma and Hodgkin disease that is progressive on
salvage therapy. Stable disease is acceptable to move forward provided it is
non-bulky.

- Active central nervous system malignancy