Patients with SCID using donors other than histocompatible siblings, Wiskott Aldrich syndrome using a histocompatible sibling donor, Wiskott Aldrich syndrome and under 5 years of age using donors other than histocompatible siblings, X-linked CD40 ligand deficiency using a histocompatible sibling donor, X-linked CD40 ligand deficiency and under 5 years of age using donors other than histocompatible siblings, other primary immunodeficiencies without manifestations of hemophagocytosis using a histocompatible sibling donor, or other primary immunodeficiencies without manifestations of hemophagocytosis and under 5 years of age using donors other than histocompatible siblings receive busulfan IV over 2 hours every 6 hours on days -9 to -6, cyclophosphamide IV on days -5 to -2, and antithymocyte globulin (ATG) twice daily on days -4 to -1. Allogeneic bone marrow or umbilical cord blood transplantation takes place on day 0. Patients receive graft versus host disease (GVHD) prophylaxis with methotrexate IV on days 1, 3, 6, and 11 and cyclosporine IV on days -3 to 50.