Pilot Trial of Allogeneic Blood or Marrow Transplantation for Primary Immunodeficiencies

* INCLUSION CRITERIA - RECIPIENT: * Patients age \>= 4 through 75 years * PID deemed to be of sufficient past severity to warrant allo BMT, by meeting the two criteria below: 1. PID as defined by identified genetic defect or, in the absence of a PID-associated genetic mutation, patients with an immune defect potentially amenable to allo BMT who meet the clinical history criteria below may be eligible upon discussion with the PI * Mutations should be confirmed in a CLIA-certified laboratory, if such testing is available. * Patients without a mutation must be deemed eligible and appropriate for allo BMT by the PI. Some patients may meet the clinical history criteria listed below, but will not be eligible if it is thought that their clinical history is due to a condition apart from an immune defect. In addition, patients with a PID of mild severity, such as those with selective IgA deficiency, may meet at least two of the clinical history criteria, but may be deemed inappropriate for allo BMT by the PI if it is felt that the risks of the procedure outweigh the severity of the disease. 2. Clinical history of at least two of the following: * Life-threatening, organ-threatening, or severely disfiguring infection * Protracted or recurrent infections requiring unusually long or repeated courses of antibiotics * Infection with an opportunistic organism * Chronic elevation in the blood (\>=2 documented elevations over a period of 6 months or longer) of a latent virus (EBV, CMV, HHV6, HHV8, etc.) * Evidence of immune dysregulation, as manifested by autoimmune disease, atopy, hemophagocytic lymphohistiocytosis/macrophage activation syndrome, granulomas, splenomegaly, or lymphadenopathy * Patients with hemophagocytic lymphohistiocytosis or macrophage activation syndrome related to an underlying lymphoma with no other clinical history suggestive of a primary immunodeficiency will not be eligible * Hypogammaglobulinemia, dysglobulinemia, or impaired response to vaccination * Hematologic malignancy or lymphoproliferative disorder * Tissue diagnosis should be confirmed by NCI Department of Pathology, if prior biopsies are available * Virus-associated solid tumor malignancy or pre-cancerous lesion * Tissue diagnosis should be confirmed by NCI Department of Pathology, if prior biopsies are available * Availability of at least one 7-8/8 (9-10/10) HLA-matched related (excluding an identical twin) or unrelated donor, or an HLA-haploidentical related donor * Consensus among the PI, key AIs, and consultants (as necessary) that correction of the patient s immune system through BMT has the potential to improve the patient s health, quality of life, and/or life expectancy, after taking into consideration the patient s existing non-hematopoietic, potentially irreversible organ dysfunction * Adequate end-organ function, as measured by: * Left ventricular ejection fraction (LVEF) \>= 40% by 2D echocardiogram (ECHO) or MUGA, or left ventricular shortening fraction \>= 20% by ECHO for patients receiving RIC or RIC-MMF, or RIC-SHORT, or LVEF \>= 30% if the patient has radiologic evidence of aortic, renal, or coronary artery vasculitis. * Pulmonary function tests: DL(co) (corrected for hemoglobin) and FEV(1) \>= 40% of predicted for the RIC, RIC-MMF, and RIC-SHORT arms; or in pediatric patients, if unable to perform pulmonary function tests, there should be no evidence of dyspnea at rest, no requirement for supplemental oxygen, and oxygen saturation \>92% on room air. Calculations will be based on the values reported in CRIS. * Bilirubin \<= 3.0 mg/dL (unless due to Gilbert s syndrome or hemolysis) for patients receiving RIC, RIC-MMF, RIC-SHORT; ALT and AST 10 x ULN for patients receiving RIC, RIC-MMF, RIC-SHORT. Patients who are above these bilirubin, ALT, or AST thresholds may be eligible for the RIC, RIC-MMF, or RIC-SHORT arms if evaluated by a hepatologist who deems the liver function test abnormalities to be potentially reversible with bone marrow transplant. * Estimated creatinine clearance of \>= 40 mL/min/1.73 m\^2, calculated using the Cockcroft-Gault equation for adults and Schwartz formula for pediatric patients, for patients with creatinine levels above the institutional upper limit of normal * Karnofsky or Lansky performance status of \>=60% or ECOG performance status of 2 or less * Ability of subject or Legally Authorized Representative (LAR) to understand and the willingness to sign a written informed consent document * Not pregnant or breastfeeding. As therapeutic agents used in this trial may be harmful to a fetus, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for at least one year post-allo BMT. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in the study, she should inform her treating physician immediately. * Disease status: Patients with malignancy are to be referred in remission for evaluation, except in cases of virus-associated malignancy who may be referred at any time. Should a patient have progressive disease or a donor becomes unavailable after enrollment, the patient will be referred back to his/her primary hematologist-oncologist for treatment. If this course of action is not in the best interest of the patient according to the clinical judgment of the PI, then the patient may receive standard treatment for the malignant disease under the current study, although this should only occur as a bridge to transplant. If under either of these settings, it becomes apparent that the patient will not be able to proceed to transplant, then he/she must come off the study. Patients receiving standard therapy will be told about the therapy, associated risks, potential benefits, alternatives to the proposed therapy, and the availability of receiving the same t