Fermat's Last Theorem is true because a non-zero integral solution to Fermat's equation with exponent bigger than 4 would give rise to an elliptic curve E over Q that is problematic in various respects. In 1986, I showed that E would violate the conjecture linking elliptic curves to modular forms. In the mid-1990s, Taylor and Wiles proved a sufficiently general variant of this modularity conjecture to obtain a contradiction to the hypothesized solution to Fermat's equation.
In my lecture, I will explain some of the ingredients of the proof, focusing especially on the modularity lifting theorems proved by Taylor and Wiles, and then later by Kisin. Making brilliant use of these theorems, and using clever new techniques, Khare and Wintenberger proved a 1987 conjecture of J-P. Serre that implies Fermat by a direct argument.