This volume contains the written versions of sixteen of the original eighteen addresses presented at the American Mathematical Society's Centennial Symposium Mathematics into the Twenty-first Century held from August 8-12, 1988. These talks, delivered at the Providence Performing Arts Center, were the principal component of the Scientific Program at the Centennial Celebration. Attendance at this meeting was unprecedented for AMS Summer Meetings with 1,949 members of the Society and a total of 2,502 in attendance including spouses, guests, etc.