**Title:** The Langlands Functoriality Conjecture: Problems in Number Theory and Representation Theory

**Abstract: **The study of *L*-functions forms an integral part of modern number theory. While this study dates back to Hecke in the 1920s, Robert Langlands, in the 1960s, began the study of *L*-functions from the point of view of automorphic representations. To an automorphic representation π of a reductive group *G*, Langlands associates an *L*-function, *L*(*s*, π ρ), where ρ is a representation of the so-called *L*-group associated to *G*. (In the simplest case, this is the group whose root datum is dual to that of *G*.) Roughly speaking, Langlands’ functoriality conjecture predicts that certain operations on the representations ρ of *L*-groups give rise to liftings of automorphic representations. In my talk, I will define these notions, give examples and motivation, and discuss recent work which addresses the functoriality conjecture and related questions in number theory.