This study will evaluate the safety and effectiveness of prednisone and methotrexate in treating severe Wegener's granulomatosis and other systemic vasculitides. These diseases involve inflammation of blood vessels (vasculitis) that may affect the brain, nerves, eyes, sinuses, lungs, kidneys, intestinal tract, skin, joints, heart and other sites. Current treatment with prednisone and the anti-cancer drug cyclophosphamide is effective, but has significant side effects and a high rate of disease recurrence. In a small number of patients with vasculitis, prednisone and methotrexate, another anti-cancer drug, have led to marked improvement, with fewer side effects than are seen with cyclophosphamide. This study will evaluate this drug combination in a larger patient population.