Inclusion Criteria:

- Patient has no existing 0-1 HLA-A, B, C, DRB1 and DQB1 matched related donor

- Acute Myeloid Leukemia:

- High risk first complete remission (CR1) as evidenced by preceding
myelodysplastic syndromes (MDS), high risk cytogenetics (for example, monosomy 5
or 7, or HR as defined by referring institution treatment protocol), >= 2 cycles
to obtain complete remission (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 (< 5% blasts) and, recovery of
peripheral blood counts with no circulating blasts, may still be eligible;
reasonable attempts must be made to obtain an adequate specimen for morphologic
assessment, including possible repeat procedures

- Acute lymphoblastic leukemia:

- 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);

- > 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 can not be obtained to
determine remission status by morphologic assessment, but have fulfilled
criteria of remission by flow cytometry (< 5% blasts) and, recovery of
peripheral blood counts with no circulating blasts, may still be eligible;
reasonable attempts must be made to obtain an adequate specimen for morphologic
assessment, including possible repeat procedures

- Chronic Myelogenous Leukemia:

- Patients in blast crisis (BC) must receive therapy and must achieve accelerated
phase (AP)/chronic phase (CP) in order to be eligible (patients who remain in BC
are not eligible);

- If in first chronic phase, patient must have failed or be intolerant to imatinib
mesylate

- Myelodysplasia (MDS):

- International Prognostic Scoring System (IPSS) Int-2 or high risk (Refractory
anemia with excess blasts [RAEB], refractory anemia with excess blasts in
transformation [RAEBt]) or refractory anemia with severe pancytopenia or high
risk cytogenetics;

- Blasts must be < 10% morphologically in representative bone marrow aspirate
(obtained < 2 weeks from enrollment)

- Lymphoblastic lymphoma, Burkitt's lymphoma, and other high-grade non-Hodgkin lymphoma
(NHL) after initial therapy if stage III/IV in first partial remission (PR1) or after
progression if stage I/II < 1 year; Stage III/IV patients are eligible after
progression in complete response (CR)/partial response (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 Patient Care Conference [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 NHL > CR2/ > PR2:

- Patients in CR2/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

- Serum creatinine =< 2.0 mg/dL (adults) and creatinine clearance > 60 ml/min
(pediatrics)

- 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 > 3 mg/dL and symptomatic biliary disease will be excluded

- Diffusing capacity of the lung for carbon monoxide corrected (DLCOcorr) > 50% normal

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

- Karnofsky score >= 70% (adults) or Lansky score >= 50% (pediatrics)

Exclusion Criteria:

- Acute leukemia in relapse (>= 5% marrow blasts by morphology)

- Active central nervous system (CNS) leukemia involvement at the time of study
enrollment (cerebrospinal fluid with > 5 white blood cells (WBC)/mm^3 AND malignant
cells on cytospin)

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

- Female patients who are pregnant or breastfeeding

- Karnofsky performance status < 70% (adults) or Lanksy score < 50% (pediatrics)

- Prior autologous or allogeneic stem cell transplant with myeloablative preparative
regimen (If =< 18 years old, prior myeloablative transplant within the last 6 months)

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

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

- Seropositive for human immunodeficiency virus (HIV)

- Consenting 5 of 6 or 6 of 6 HLA-matched related donor available

- Unable to provide informed consent

- Use of any other experimental drug within 28 days of baseline