Abstract. This talk will begin with a survey-level discussion of Lie groups, their unitary representations, and why one might be interested in them. We will then delve a bit into the details of how Lie groups were classified, and how the five exceptional groups were seen to exist. We'll describe a conjectural structure to the class of unitary representations, and explain an approach to proving parts of these conjectures. Time permitting we'll report on some recent results with Stephen Miller of Rutgers on this problem.