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.