This study is to discover whether children with severe combined immunodeficiency disease (SCID) or other primary immunodeficiency disorder (PID) for which no satisfactory treatment other than stem cell transplantation (SCT) exists can be safely and effectively transplanted from HLA mismatched (up to one haplotype) related donors or unrelated matched or mismatched (up to one antigen) donors, when leukocytolytic monoclonal antibodies (MAb) and Fludarabine are the sole conditioning agents. Three monoclonal antibodies will be used in combination. Two of them are rat IgG1 (immunoglobulin G1) antibodies directed against two contiguous epitopes on the CD45 (common leucocyte) antigen. They have been safely administered as part of the conditioning regimen for 12 patients receiving allografts (HLA matched and mismatched) at this center. They produce a transient depletion of >90% circulating leucocytes. The third MAb is Campath 1H, a humanized rat anti-CD52 MAb. Campath 1H, Alemtuzumab, has been licensed to treat B-cell chronic lymphocytic leukemia (B-CLL) and more recently has been safely given at this and other centers as part of a sub-ablative conditioning regimen to patients with malignant disease. Because these MAb produce both profound immunosuppression and significant, though transient, myelodestruction we believe they may be useful as the sole conditioning regimen in patients with SCID, in whom the use of conventional chemotherapeutic agents for conditioning may produce or aggravate unacceptable and even lethal short term toxicity. We anticipate MAb mediated subablative conditioning will permit engraftment in a high percentage of these patients with little or no immediate or long term toxicity. Campath IH persists in vivo for several days after administration and so will be present over the transplant period to deplete donor T cells as partial GvHD prophylaxis. Additional Graft versus Host Disease (GvHD) prophylaxis may be provided by administration of FK506.