Sirolimus/Tacrolimus Versus Tacrolimus/Methotrexate for Preventing Graft-Versus-Host Disease (GVHD) (BMT CTN 0402)

Inclusion Criteria: * 6/6 HLA-matched sibling, defined by Class I (HLA-A and B) serologic typing (or higher resolution) and Class II (HLA-DRBI) molecular typing, who is willing to donate peripheral blood stem cells, and meets institutional criteria for stem cell donation. The donor must be medically eligible to donate stem cells, according to individual transplant center criteria. Pediatric patients for whom a pediatric sibling donor is not anticipated to be a suitable leukapheresis candidate are not eligible. * Karnofsky performance status of at least 70% or Lansky performance status of at least 70% for participants less than 16 years old * For participants less than 18 years old, willing and able to take oral medications, per the treating physician's recommendations Exclusion Criteria: * Prior allogeneic or autologous transplant using any hematopoietic stem cell source * Seropositive for the human immunodeficiency virus (HIV) * Uncontrolled bacterial, viral, or fungal infection (currently taking medication and progression of clinical symptoms) * Pregnant (positive serum human chorionic gonadotropin \[β-HCG\] test) or breastfeeding within 4 weeks of study entry * Kidney function: serum creatinine outside the normal range for age, or measured creatinine clearance less than 50 mL/min/1.72m\^2 within 4 weeks of study entry * Liver function: most recent direct bilirubin, alanine aminotransferase (ALT), or aspartate aminotransferase (AST) greater than two times the upper limit of normal within 4 weeks of study entry * Lung disease: in adults, forced vital capacity (FVC) or forced expiratory volume in one second (FEV1) less than 60% of predicted value (corrected for hemoglobin); in children, overt hypoxemia, as measured by an oxygen saturation of less than 92% within 4 weeks of study entry * Cardiac ejection fraction of less than 45% in adults and children, or less than 26% shortening fraction in children within 4 weeks of study entry * Cholesterol level greater than 500 mg/dL or triglyceride level greater than 500 mg/dL while being treated, or not on appropriate lipid-lowering therapy within 4 weeks of study entry * Prior history of allergy to sirolimus * Requires voriconazole at time of study entry * Currently receiving another investigational drug unless cleared by the protocol officer or protocol chair * Participants with a history of cancer, other than resected basal cell carcinoma or treated carcinoma in-situ. Cancer treated with curative intent for more than 5 years previously will be allowed. Cancer treated with curative intent for less than 5 years previously will not be allowed unless approved by the protocol officer or protocol chair.