When Andrew Wiles announced in June 1993 that he had proved Fermat's Last Theorem, mathematicians and nonexperts alike applauded his achievement. The proof was finally completed in November 1994, when Wiles and Richard Taylor filled a technical gap that had arisen. In this lecture, delivered two months after Wiles's historic announcement, Barry Mazur outlines the main ideas in this ground-breaking work. Mazur sets the context for the problem posed in Fermat's Last Theorem by discussing the ABC-Conjecture. He then focuses on three concepts and the key role they play in Wiles's work: elliptic curves, Galois representations, and modular forms. An introduction by Mazur, prepared in March 1995, is included on this DVD. Specialists and nonspecialists alike will appreciate this lucid and insightful exposition. It would be accessible to students with background in number theory and algebra.