Study of Gene Modified Donor T-cells Following TCR Alpha Beta Positive Depleted Stem Cell Transplant

Inclusion Criteria: 1. Age \> 1 month and \< 26 years 2. Life expectancy \> 10 weeks 3. Subjects deemed eligible for allogeneic stem cell transplantation. 4. Subjects with life-threatening hematological malignancies (high-risk ALL in 1st CR, ALL in 2nd or subsequent CR, AML in 1st CR, AML in 2nd or subsequent CR, myelodysplastic syndromes, non-Hodgkin lymphomas in 2nd or subsequent CR, other hematologic malignancies eligible for stem cell transplantation per institutional standard); 5. Non-malignant disorders amenable to cure by an allograft: 1. primary immune deficiencies, 2. severe aplastic anemia not responding to immune suppressive therapy, 3. osteopetrosis, 4. hemoglobinopathies, (thalassemias, and sickle cell anemia, and Diamond-Blackfan anemia among others) 5. congenital/hereditary cytopenia, including Fanconi Anemia before any clonal malignant evolution (MDS, AML) Note: Subjects will be eligible if they meet either item 4 OR item 5. 6. Lack of suitable conventional donor (HLA identical sibling or HLA phenotypically identical relative or 10/10 unrelated donor evaluated using high resolution molecular typing) or presence of rapidly progressive disease not permitting time to identify an unrelated donor 7. A minimum genotypic identical match of 5/ 10 is required. 8. The donor and recipient must be identical, as determined by high resolution typing, at least one allele of each of the following genetic loci: HLA-A, HLA-B, HLA-Cw, HLA- DRB1 and HLA-DQB1. 9. Lansky/Karnofsky score \> 50 10. Signed written informed consent Exclusion Criteria: 1. Greater than Grade II acute GVHD or chronic extensive GVHD due to a previous allograft at the time of inclusion 2. Subject receiving an immunosuppressive treatment for GVHD treatment due to a previous allograft at the time of inclusion 3. Dysfunction of liver (ALT/AST \> 5 times normal value, or bilirubin \> 3 times normal value), or of renal function (creatinine clearance \< 30 mL / min) 4. Severe cardiovascular disease (arrhythmias requiring chronic treatment, congestive heart failure or left ventricular ejection fraction \< 40%) 5. Current active infectious disease (including positive HIV serology or viral RNA) 6. Serious concurrent uncontrolled medical disorder 7. Pregnant or breastfeeding subject 8. For subjects who have received more than 1 x 10E5 alpha/beta T cells/kg with the graft infusion the clinical trial site must contact the sponsor for approval to be eligible to receive BPX-501 infusion.