Haploidentical Allogeneic Transplant With Post-transplant Infusion of Regulatory T-cells

Inclusion Criteria RECIPIENT * Histopathologically-confirmed: * Acute leukemia (in first remission with poor risk factors and molecular prognosis) * Acute myelogenous leukemia (AML) with -5,-7, t (6;9), tri8, -11 * Acute lymphoblastic leukemia (ALL) with Ph+ t (9;22), t (4;22), (q34;q11) * Acute leukemia with refractory disease or \> Complete Remission (CR) 1 * Chronic myelogenous leukemia (CML) (accelerated, blast or second chronic phase) * Myelodysplastic syndrome (in high and high intermediate risk categories) * Non-Hodgkin's lymphoma (NHL) with poor risk features and not suitable for autologous transplantation * Refractory Chronic lymphocytic leukemia (CLL) * At least 21 days from the end of most recent prior therapy to start of the transplant conditioning regimen * Must be \< 60 years old at time of registration. * Karnofsky Performance Status (KPS) \> 70% Must have related donor who is: * Genotypically human leukocyte antigen (HLA) -A, B,C and DR beta 1 (DRB1), DQ loci haploidentical to the recipient (but differing for 2 to 3 HLA alleles on the unshared haplotype in the graft-versus-host disease (GvHD) direction) * No HLA-matched sibling or matched-unrelated donor is identified. * Adequate cardiac and pulmonary function (left ventricular ejection fraction (LVEF) \> 45%, diffusing capacity of the lungs for carbon monoxide (DLCO) \>50% corrected for hemoglobin) * Serum creatinine \< 1.5 mg/dL OR Creatinine clearance \> 50 mL/min for those above serum creatinine at least 1.5 mg/dL * Serum bilirubin \< 2.0 mg/dL * Alanine transaminase (ALT) \< 2x upper normal limit (ULN) (unless secondary to disease) * No prior myeloablative therapy or hematopoietic cell transplantation DONOR: * Age ≤ 70 years * Weight ≥ 25 kg. * Medical history and physical examination confirm good health status as defined by institutional standards * Seronegative for HIV Ag within 30 days of apheresis collection for: * Hepatitis B surface antigen (sAg) or polymerase chain reaction (PCR) + * Hepatitis C ab or PCR+ * Genotypically haploidentical as determined by HLA typing * Female donors (child-bearing potential) must have a negative serum or urine beta-human chorionic gonadotropin (HCG) test within 3 weeks of mobilization * Capable of undergoing leukapheresis * Has adequate venous access * Willing to undergo insertion of a central catheter if leukapheresis via peripheral vein is inadequate * Capable of agreeing to second donation of peripheral blood progenitor cell (PBPC) (or a bone marrow harvest) should the patient fail to demonstrate sustained engraftment following the transplant * Institutional review board (IRB)-approved consent form signed by donor or legal guardian \> 18 years of age Donor Selection in the priority order: * Recipient's biological mother preferred, if available * Other available haploidentical donors will be selected based upon the presence of natural killer (NK) alloreactivity between donor and recipient by high-resolution HLA typing of the C locus. An NK-alloreactive donor will be preferentially chosen. Recipients lacking a killer immunoglobulin-like receptor (KIR)-ligand present in the donor along with the corresponding KIR defines "NK alloreactivity". * If more than one NK-alloreactive donor is available, preference is to cytomegalovirus (CMV)-seronegative donor Exclusion Criteria RECIPIENT: * Suitable candidate for autologous transplantation or allogeneic transplantation with an available matched-related or matched-unrelated donor * Seropositive for: * HIV ab * Hepatitis B sAg or PCR+ * Hepatitis C ab or PCR+ * History of invasive Aspergillosis * Any active, uncontrolled bacterial, viral or fungal infection * Uncontrolled central nervous system (CNS) disease involvement * Lactating female DONOR: * Evidence of active infection or viral hepatitis * Factors of increased risk for complications from leukapheresis or granulocyte-colony stimulating factor (G-CSF) therapy * Lactating female * HIV-positive