Over the last 15 years, there have been tremendous advances in our understanding of the connections among modular forms, Galois representations and algebraic varieties. Undoubtedly, the most spectacular development in this subject was the proof of Fermat's Last Theorem, which was completed in 1994. Beginning in the late 1960s, J-P. Serre proposed links of various kinds between modular forms and representations of Galois groups. In 1987, Serre wrote a seminal article that included precise conjectures relating mod p Galois representations and mod p modular forms. These conjectures were so powerful and general that they were inaccessible by then-current methods. Amazingly, these conjectures have been proved over the last two years, with the final step being contributed only several months ago. The main ideas are due to Khare and Wintenberger, with major contributions from Kisin and others. My talk will explain the history of the conjectures and some elements of the ingenious proof.