Biparental HLA Haplotype Disparate T-cell Depleted Transplants for Patients Lacking an HLACompatible Donor

Inclusion Criteria: * Malignant conditions for which CD34+ selected, T-cell depleted allogeneic hematopoietic stem cell transplantation is indicated such as: AML in 1st remission - for patients whose AML does not have 'good risk' cytogenetic features (i.e. t 8;21, t15;17, inv 16). * Secondary AML in 1st remission * AML in 1st relapse or \> 2nd remission * ALL/LL in 1st remission clinical or molecular features indicating a high risk for relapse; or ALL \> 2nd remission * CML failing to respond to or not tolerating Imatinib, dasatinib, or nilotinib in first chronic phase of disease; or CML in accelerated phase or second chronic phase. * Non-Hodgkins lymphoma with chemoresponsive disease in any of the following categories: a) intermediate or high grade lymphomas who have failed to achieve a first CR or have relapsed following a 1st remission who are not candidates for autologous transplants. * any NHL in remission which is considered not curable with chemotherapy alone and not eligible/appropriate for autologous transplant. Myelodysplastic syndrome (MDS): RA/RCMD with high risk cytogenetic features or transfusion dependence, RAEB-1 and RAEB-2 and Acute myelogenous leukemia (AML) evolved from MDS, who are not eligible for transplantation under protocol IRB 08-008. * Chronic myelomonocytic leukemia: CMML-1 and CMML-2. * Other rare lethal disorders of Hematopoiesis and Lymphopoiesis for which a T-cell depleted transplant is indicated (e.g. hemophagocytic lymphohistiocytosis; refractory aplastic anemia or conjugated cytopenias; non-SCID lethal genetic immunodeficiencies such as Wiskott Aldrich Syndrome, CD40 ligand deficiency, ALPS). * Patients may be of either gender and of any racial or ethnic background. * Patients must have a Karnofsky (adult) or Lansky (pediatric) Performance Status \> 70%. * Patients must have adequate organ function measured by: Cardiac: asymptomatic or if symptomatic then LVEF at rest must be \> 50% and must improve with exercise. * Hepatic: \< 3x ULN ALT and \< 2.0x ULN total serum bilirubin, unless there is congenital benign hyperbilirubinemia. * Renal: serum creatinine \<1.2 mg/dl or if serum creatinine is outside the normal range, then CrCl \> 40 ml/min (measured or calculated/estimated) * Pulmonary: asymptomatic or if symptomatic, DLCO \> 50% of predicted (corrected for hemoglobin) * Each patient must be willing to participate as a research subject and must sign an informed consent form. Exclusion Criteria: * Female patients who are pregnant or breast-feeding * Uncontrolled viral, bacterial or fungal infection * Patient seropositive for HIV-I/II; HTLV -I/II * Presence of leukemia in the CNS. Donor Inclusion Criteria: * Each donor must meet criteria outlined by institutional policies * Donor must have adequate peripheral venous catheter access for leukapheresis or must agree to placement of a central catheter. Donor Exclusion Criteria: * Evidence of active infection (including urinary tract infection, or upper respiratory tract infection), viral hepatitis exposure (on screening), unless only HBS Ab+ and HBV DNA negative, or serologic evidence of exposure or infection with HIV-I/II or HTLV-I/II * If donors do not meet institutional guidelines, exclusion will be considered.