Patient enrollment is staged to allow study physicians to closely monitor patients for signs of disseminated varicella reactions or graft rejection. Initially only 1 patient will be enrolled in the study. If the first patient reaches Week 8 without a severe adverse reaction, 3 study centers will then enroll 3 additional patients. If 8 weeks later these 3 patients have had no severe adverse reactions, the same 3 study centers will enroll 3 more patients. At the end of this period, having ascertained the safety of the vaccine in the first 7 patients, the study will be opened to the remaining centers. Patients receive 2 doses of Varivax 6 to 8 weeks apart. Each week for 6 to 8 weeks after the first vaccine dose, the patient undergoes venipuncture and clinical assessment to characterize renal graft and liver function and identify any signs of varicella infection. Additional telephone follow-up occurs on Day 4 and twice weekly thereafter. Parents or guardians monitor the patient for evidence of rash or fever and immediately report any rashes or fevers to study physicians. If, after 6 to 8 weeks, the patient demonstrates no severe reactions to the vaccine and requires no antiviral therapy, the patient receives the second vaccine dose. The patient again receives weekly on-site and telephone follow-up for 6 weeks. Other visits occur 9 weeks and 14 weeks after the second vaccine dose and 1 year after the first vaccine dose. At these visits the patient undergoes venipuncture and clinical assessment to identify potential rejection events or varicella infection and to characterize VZV antibody responses and cytokine changes in response to the vaccine.