Hemoglobinopathies, such as sickle cell disease and thalassemia major, are genetic diseases associated with significant morbidity and premature death. Allogeneic bone marrow transplantation (BMT) is the only potential cure for severe hemoglobinopathies. Typical regimens have used high doses of chemotherapy or chemo-radiotherapy to ablate recipient hematopoiesis and to prevent graft rejection. The widespread use of this treatment has been limited by toxicity, risk of end-organ damage, and donor availability. This study will use a nonmyeloablative regimen of fludarabine and busulfan to attempt to generate consistent engraftment with donor hematopoietic stem cells in patients with severe hemoglobinopathy.