the goal of the talk is to give some idea of what the all-important langlands program is all about, and it is aimed at a general audience with no or limited experience in number theory. for $gl(2)$ we will illustrate these ideas by discussing the case of elliptic curves and modular forms, which led to a proof of fermat's last theorem! we will then try to give a flavor of what's expected for $gl(n)$, and point to how the so-called p-adic langlands program was envisaged. this ties up with the number theory topics class offered in the spring quarter.