This is a phase I/II study to determine the safety and tolerance of the protease inhibitor indinavir (MK-0639), alone and then in combination with HIV reverse transcriptase inhibitor therapy in children with HIV infection. Indinavir sulfate (the capsule formulation) has been shown to have potent antiviral activity and an acceptable safety profile in adults. HIV-infected children who have not received prior antiretroviral therapy, and children who have become refractory to prior therapy, or who have experienced toxicity to prior therapy, will be included. In addition, we will explore viral and CD4 cell kinetics before starting therapy and following exposure to antiretroviral agents.