Inclusion Criteria:

1. Diagnosed with one of the following diseases:
2. Acute myelogenous leukemia (AML) in induction failure, relapse, past first remission, or CR1 considered at risk for relapse
3. Myelodysplastic syndromes with International Prognostic Scoring System score (IPSS score) >/= 2 or myelodysplasia that has not responded to chemotherapy
4. Biphenotypic leukemia
5. Acute lymphocytic leukemia with induction failure, first complete remission with high risk cytogenetics (e.g. Philadelphia positive chromosome, t(4:11) Remission requiring more than 2 chemotherapy to achieve remission, or any stage beyond CR1
6. Chronic Myelogenous Leukemia (CML): second chronic phase, accelerated phase or blast crises with less than 10% blasts in the bone marrow, or CR1 and resistance to Gleevec or other tyrosine kinase inhibitors
7. Non-Hodgkin's Lymphoma - induction failures, second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant)
8. Hodgkin's disease - induction failure, second or later complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant).
9. Chronic Lymphocytic Leukemia that has failed induction therapy or Rai Stages 2-4
10. Related or unrelated donor which is HLA-matched or mismatched in 1 HLA A, B, C, DR, or DQ locus is acceptable (i.e. >/= 9/10 matched related or unrelated donor, matched with molecular high-resolution technique per current std. for BMT program). Cord blood units must match patient at 4, 5, or 6/6 HLA class 1 serological & II molecular antigens with a min. of 2 x 10e7 TNC/kg recipient weight in the pre-thawed fraction. For patient lacking a matched related or unrelated donor or acceptable cord blood unit(s), a related haploidentical donor (</= 7/8 allele matched at A, B, C, DR loci) may be used.
11. Age </= 60 years.
12. Lansky performance score >/= 50% for patients </= 16 years of age, or Zubrod performance status score of 0-2 for patients > 16 years of age.
13. Cardiac function - left ventricular ejection fraction >/= 40%.
14. Pulmonary function - diffusion capacity of at least 50% predicted. Children unable to perform pulmonary function tests (e.g. less than 7 years old) pulse oximetry of >/= 92% on room air.
15. Serum creatinine < 1.6 mg/dL or creatinine clearance >/= 50 ml/min.
16. SGPT </= 200 IU/mL, serum bilirubin < 1.5 x normal.
17. Written informed consent and assent as is age appropriate.
18. No active infection.

Exclusion Criteria:

1. Pregnancy in women of child bearing potential (pregnancy test performed within 2 weeks of study entry).
2. HIV positive (highly immunosuppressive treatment)
3. Active CNS leukemia
4. Chronic or active Hepatitis B or Hepatitis C. If questions about liver health discuss with PI and strongly consider liver biopsy.