Elliptic curves appear prominently in Wiles's resolution of Fermat's Last Theorem, the Birch and Swinnerton-Dyer Conjecture (a Millennium Problem), and Lenstra's factoring algorithm, and they are also important in modern cryptography. In this talk, we will give a gentle introduction to elliptic curves and then explain some of these applications without assuming any background in algebraic geometry or number theory. Sponsored by the GSA.