This is a single arm pilot study for patients using α/β T cell-depleted peripheral Stem Cell Transplantation (PSCT) in with alternative donor sources with hematologic malignancies receiving alternative donor (unrelated or partially matched related) mobilized peripheral stem cells (PSCs) using the CliniMACS system for T cell depletion plus CD19+ B cell depletion to determine efficacy as determined by engraftment and graft versus host disease (GVHD), and one year leukemia free survival.

Inclusion Criteria:

1. Leukemias/lymphomas

   • Acute myeloid leukemia, primary or secondary Disease status: remission or <10% bone marrow blasts
   • Myelodysplasia
   • Acute lymphoblastic leukemia Disease status: in hematologic remission
   • Chronic myelogenous leukemia Disease status: chronic phase, accelerated phase or blast crisis now in second chronic phase.
   • Mixed lineage or biphenotypic acute leukemia
   • Lymphoblastic lymphoma Disease status: remission
   • Burkitt's lymphoma/leukemia: Disease status- in remission

2. Acceptable organ status clearance

3. No active untreated infection. Patients with possible fungal infections must have had at least 2 weeks of appropriate anti-fungal antibiotics and be asymptomatic

4. Signed informed consent.

5. Negative pregnancy test for females of childbearing potential.

Exclusion Criteria:

1. Patients who do not meet disease, organ or infectious criteria.
2. No suitable donor
3. Received previous allogeneic transplant

Donor must be either unrelated, mismatched up to one antigen/one antigen and one allele or a related, HLA match other than a sibling.