Trial discontinued prior to starting Phase II portion due to low accrual.

Open label, dose finding trial to assess the efficacy of Treg/Tcon addback to partially matched related donor stem cells. The maximum tolerated dose will be established using 3 subjects per dose level, with an expansion cohort at the maximum tolerated dose.

Inclusion Criteria:

1. Age 0-70 years 2. Karnofsky or Lansky Performance status >70% 3. High risk hematologic malignancy 4. Acute myeloid leukemia (AML) with one or more of the following criteria: 4a.Poor risk cytogenetics, including -5, 5q-, -7, 7q-, t(9;22); complex cytogenetics (>3 abnormalities); or normal cytogenetics with Flt3 internal tandem duplication (ITD), in first or subsequent complete remission (CR).

4b. Relapsed or primary refractory AML with <10% blasts in the peripheral blood.

4c. Subjects in first complete remission (CR1) who required two cycles of induction to achieve remission may be included at the discretion of the treating physician.

4d. Standard risk or intermediate risk cytogenetics in second or subsequent CR (enrolled at the discretion of the treating physician).

5. Acute lymphoblastic leukemia (ALL) with one of the following criteria: 5a. Second or subsequent CR 5b. Any partial remission (PR) (no circulating blasts) 5c. High-risk ALL in first CR including (Ph+, t(4:11), complex karyotype, hypodiploidy (<44 chromosomes), or positive minimal residual disease (MRD) after induction 6. Myelodysplasia, intermediate -2 (score 1.5-2.0) or high risk (score >2.5) by the International Prognostic Score System.

7. Myeloproliferative Disorders (include chronic myelomonocytic leukemia (CMML), agnogenic myeloid metaplasia (AMM) or Idiopathic Myelofibrosis, and JMML) with excess blasts (>5%) 8. Chronic myeloid leukemia (CML) with one of the following criteria: 8a. Second or subsequent chronic phase 8b. Accelerated phase 8c. blast crisis 9. Non-Hodgkin's lymphoma (NHL) meeting one of the following criteria: 9a. Relapse after autologous stem cell transplantation with evidence of responsive disease.

9b. Subject with chemosensitive relapse who have no option for autologous stem cell transplantation due to blood or marrow involvement or failure to mobilize autologous stem cells or are not considered eligible for autologous transplant by their treating physician.

9c. Hodgkin's Lymphoma: relapse after autologous hematopoietic cell transplantation (HCT), chemo-refractory disease 9d. Multiple myeloma: per National Comprehensive Cancer Network (NCCN) guidelines. Updated annually at: www.nccn.org 10. No suitable human leukocyte antigen (HLA)-identical sibling donor. 11. No identified 8/8 (based upon A, B, C, DR beta 1 (DRB1) loci) allele matched unrelated donor, or unable to wait sufficient time to procure a 8/8 allele matched unrelated donor 12. Available HLA 3-5/6 matched genotypically haploidentical partially matched related donor 13. Female subjects must be surgically sterile, postmenopausal (minimum 1 year without menses), or agree to use approved form of contraception from the time of signing the informed consent form through Day +100. Male subjects must also agree to use an approved form of birth control for either themselves or their partner, as appropriate, from the time of signing the informed consent form through Day +100.

14. Able to provide informed consent and have signed an approved consent form that conforms to federal and institutional guidelines.

Exclusion Criteria:

1. Available HLA identical matched sibling donor (unless having failed a prior allogeneic transplant from an HLA identical matched sibling)

2. Recipient HLA antibodies against donor HLA

3. Any of the following organ dysfunctions:

   1. Cardiac- left ventricular ejection fraction <40%, symptomatic coronary artery disease, or uncontrolled arrhythmias
   2. Pulmonary- forced expiratory volume at one second (FEV1) or diffusion capacity of lung for carbon monoxide (DLco)<40% or need for use of supplemental oxygen
   3. Renal- calculated or measured glomerular filtration rate (GFR) <30 ml/min, dialysis requirement, or prior renal transplant
   4. Hepatic- bilirubin > 2.0, alanine aminotransferase (ALT) > 2.5 X upper limit of normal (ULN), cirrhosis

4. Subjects with active or uncontrolled bacterial, viral, or fungal infections requiring systemic therapy.

5. Subjects who have tested positive for HIV.

6. Pregnant women, nursing mothers or women of child-bearing potential who are unwilling to use medically accepted methods of contraception.