Topiramate is a medication that is widely used for the treatment of adult and pediatric patients with seizures and has been shown in preliminary studies to be effective for migraine prevention in adults. This is an outpatient, randomized, double-blind, placebo-controlled study to confirm preliminary studies of the effectiveness of topiramate in the prevention of migraine attacks. The study is composed of 4 phases: pretreatment, double-blind treatment for 20 weeks, a blinded transition, and an open-label extension. During the pretreatment phase patients discontinue all medication for migraine prevention and keep a daily record of headache information in a diary. Patients with 3 to 8 migraines, but not more than 15 headache days, during the pretreatment phase continue in the double-blind treatment phase. In the 20-week double-blind treatment phase, oral topiramate tablets (or placebo) are taken daily beginning at 25 mg once daily for 1 week, increasing to twice daily doses of up to a maximum of 8 tablets (200 mg) per day during the 8-week titration period, and maintained at that dose during the 12-week maintenance period. Patients who successfully complete the double-blind therapy phase may continue in an open-label extension (32 weeks), during which the study medication (topiramate or placebo) will be decreased and simultaneously topiramate (open-label) will be administered at increasing doses. During the study, patients will maintain headache and medication records to document the following: occurrence and duration of headaches; severity of headache pain; associated symptoms, such as nausea, vomiting, photophobia (avoidance and dread of light), phonophobia (fear of sound); and medication taken to relieve headache pain or symptoms. The primary measure of effectiveness is the percent reduction in the frequency of monthly migraine episodes during the entire double-blind treatment phase compared with the pretreatment phase. Other assessments of effectiveness include the percent of patients responding to treatment (>= 50% reduction in average monthly migraine attacks) during the double-blind treatment phase compared with the pretreatment phase, the reduction in number of migraine days/month during treatment, and the reduction in severity and duration of migraines during treatment. Safety assessments include the incidence of adverse events throughout the study, and measurement of vital signs (pulse, blood pressure, body weight), physical examinations, and clinical laboratory tests (hematology, biochemistry, and urinalysis) at specified intervals. The study hypothesis is that the decrease in the mean monthly migraine rate is greater in the topiramate group than in the placebo group.