T-cell Depleted Alternative Donor Transplantation

Inclusion Criteria: * Age \< 30 years * Patient must have a malignant or non-malignant disease that can benefit from alternative stem cell transplantation. Examples include acute and chronic leukemias, myelodysplastic syndrome, lymphoma, severe acquired and congenital cytopenias, white and red blood cell abnormalities, and immunodeficiencies. * Patients with acute lymphoblastic leukemia must be in morphological remission (\< 5% blasts) at the time of transplant. Patients with acute non-lymphocytic leukemia will preferably be in morphologic remission but may be enrolled when aplastic after chemotherapy or with \< 20% blasts. Patients with lymphoma must be in complete or close to complete remission (if residual adenopathy, PET scan must be negative or only have slight uptake, eg. SUV \< 2). * Patients must lack a healthy HLA-identical related donor of at least one year of age. * Patient must have a mismatched related or an unrelated donor who is: 1. Able to receive G-CSF and undergo apheresis either through placement of catheters in antecubital veins or a temporary central venous catheter, 2. Healthy, 3. Willing, 4. For recipients of an unrelated donor transplant, recipient eligibility will be restricted as follows if in the judgment of the recipients' transplant physician, the recipient cannot receive a transplant with combined positive and negative fractions as described in Section 6.1.3.2 or an unmanipulated PBSC product. 5. Meets eligibility criteria for donors. * If only one mismatched related relative is available, an acceptable unrelated donor must be identified as a backup. * Patient or authorized guardian must sign informed consent for this study. Exclusion Criteria: * Patient with an anticipated life expectancy of \< 1 month * Active infectious hepatitis or CMV infection * HIV or HTLV-I/II infection * Serious infection (bacterial, fungal, viral) within the last 4 weeks * Cardiac ejection fraction \< 45%; can be lower if patient is not in clinical cardiac failure and a reduced intensity conditioning regimen is used. * Creatinine clearance \<60 ml/min/1.72 m2; can be lower if a reduced intensity conditioning regimen is used. * Pulmonary diffusion capacity (adjusted for Hgb), FEV1, or FVC \<60% of predicted or O2 sat \< 94% if unable to perform PFTs; can be lower if a reduced intensity conditioning regimen is used. * Serum ALT \> 3 x upper limit of normal (can be up to 5 x upper limit of normal if a reduced intensity conditioning regimen is used) or bilirubin \> 2. The bilirubin criteria for sickle cell disease patients is direct bilirubin \>2x upper limit of normal. * Performance score (Lansky/Karnofsky) \< 50 * Any condition that compromises compliance with the procedures of this protocol, as judged by the principal investigator.