Patient Inclusion Criteria:

1. Patient age 0.5-75 years
2. Absence of a suitable related or unrelated bone marrow donor who is molecularly matched at HLA-A, B, Cw, DRB1, and DQB1.
3. Absence of a suitable partially HLA-mismatched (haploidentical), first-degree related donor. Donors who are homozygous for the CCR5delta32 polymorphism are given preference.
4. Eligible diagnoses:

1. Relapsed or refractory acute leukemia in second or subsequent remission, with remission defined as \<5% bone marrow blasts morphologically
2. Poor-risk acute leukemia in first remission, with remission defined as \<5% bone marrow blasts morphologically:

* AML with at least one of the following:

* AML arising from MDS or a myeloproliferative disorder, or secondary AML
* Presence of Flt3 internal tandem duplications
* Poor-risk cytogenetics: Complex karyotype \[\> 3 abnormalities\], inv(3), t(3;3), t(6;9), MLL rearrangement with the exception of t(9;11), or abnormalities of chromosome 5 or 7
* Primary refractory disease
* ALL (leukemia and/or lymphoma) with at least one of the following:

* Adverse cytogenetics such as t(9;22), t(1;19), t(4;11), or MLL rearrangement
* Clear evidence of hypodiploidy
* Primary refractory disease
* Biphenotypic leukemia
3. MDS with at least one of the following poor-risk features:

* Poor-risk cytogenetics (7/7q minus or complex cytogenetics)
* IPSS score of INT-2 or greater
* Treatment-related MDS
* MDS diagnosed before age 21 years
* Progression on or lack of response to standard DNA-methyltransferase inhibitor therapy
* Life-threatening cytopenias, including those generally requiring greater than weekly transfusions
4. Interferon- or imatinib-refractory CML in first chronic phase, or non-blast crisis CML beyond first chronic phase.
5. Philadelphia chromosome negative myeloproliferative disease.
6. Chronic myelomonocytic leukemia.
7. Juvenile myelomonocytic leukemia.
8. Low-grade non-Hodgkin lymphoma (including SLL and CLL) or plasma cell neoplasm that has:

* progressed after at least two prior therapies (excluding single agent rituximab and single agent steroids), or
* in the case of lymphoma undergone histologic conversion;
* patients with transformed lymphomas must have stable disease or better.
9. Poor-risk CLL or SLL as follows:

* 11q deletion disease that has progressed after a combination chemotherapy regimen,
* 17p deletion disease,
* or histologic conversion;
* patients with transformed lymphomas must have stable disease or better.
10. Aggressive non-Hodgkin lymphoma as follows, provided there is stable disease or better to last therapy:

* NK or NK-T cell lymphoma, hepatosplenic T-cell lymphoma, or subcutaneous panniculitic T-cell lymphoma, blastic/ blastoid variant of mantle cell lymphoma
* Hodgkin or aggressive non Hodgkin lymphoma that has failed at least one multiagent regimen, and the patient is either ineligible for autologous BMT or autologous BMT is not recommended.
* Eligible subtypes of aggressive non-Hodgkin lymphoma include:

* mantle cell lymphoma
* follicular grade 3 lymphoma
* diffuse large B-cell lymphoma or its subtypes, excluding primary CNS lymphoma
* primary mediastinal large B-cell lymphoma
* large B-cell lymphoma, unspecified
* anaplastic large cell lymphoma, excluding skin-only disease
* Burkitt's lymphoma or atypical Burkitt's lymphoma (high-grade B-cell lymphoma, unclassifiable, with features intermediate between diffuse large B-cell lymphoma and Burkitt's), in complete remission
5. Patients with CLL, SLL, or prolymphocytic leukemia must have \< 20% bone marrow involvement by malignancy (to lower risk of graft rejection).
6. One of the following, in order to lower risk of graft rejection:

* Cytotoxic chemotherapy, an adequate course of 5-azacitidine or decitabine, or alemtuzumab within 3 months prior to start of conditioning; or
* Previous BMT within 6 months prior to start of conditioning.

NOTE: Patients who have received treatment outside of these windows may be eligible if it is deemed sufficient to reduce graft rejection risk; this will be decided on a case-by-case basis by the PI or co-PI.
7. Any previous BMT must have occurred at least 3 months prior to start of conditioning.
8. Adequate end-organ function as measured by:

1. Left ventricular ejection fraction greater than or equal to 35%, or shortening fraction \> 25%, unless cleared by a cardiologist
2. Bilirubin ≤ 3.0 mg/dL (unless due to Gilbert's syndrome or hemolysis), and ALT and AST \< 5 x ULN
3. FEV1 and FVC \> 40% of predicted; or in pediatric patients, if unable to perform pulmonary function tests due to young age, oxygen saturation \>92% on room air
9. ECOG performance status \< 2 or Karnofsky or Lansky score \> 60

Patient Exclusion Criteria:

* Not pregnant or breast-feeding.
* No uncontrolled bacterial, viral, or fungal infection.

* Note: HIV-infected patients are potentially eligible. Eligibility of HIV-infected patients will be determined on a case-by-case basis.
* No previous allogeneic BMT (syngeneic BMT permissible).
* Active extramedullary leukemia or known active CNS involvement by malignancy. Such disease treated into remission is permitted.

Donor Inclusion Criteria:

1. Potential donors consist of:

* Unrelated donors
* Second-degree relatives
* First cousins
2. The donor and recipient must be identical at at least 5 HLA alleles based on high resolution typing of HLA-A, -B, -Cw, -DRB1, and -DQB1, with at least one allele matched for a HLA class I gene (HLA-A, -B, or -Cw) and at least one allele matched for a class II gene (HLA-DRB1 or -DQB1).
3. Meets institutional selection criteria and medically fit to donate. 4 . Lack of recipient anti-donor HLA antibody. Note: In some instances, low level, non-cytotoxic HLA specific antibodies may be permissible if they are found to be at a level well below that detectable by flow cytometry. This will be decided on a case-by-case basis by the PI and one of the immunogenetics directors. Pheresis to reduce anti-HLA antibodies is permissible; however eligibility to proceed with the transplant regimen would be contingent upon the result.

Donor Exclusion Criteria:

* Donor must not be HLA identical to the recipient.
* Has not donated blood products to recipient.