Avi Wigderson, of the Institute for Advanced Study, surveys five decades of work on constructing expanding Cayley graphs of finite groups. Speaking in the Computer Science/Discrete Mathematics Seminar II at IAS, he walks through the main techniques for building these expander graphs and the mathematical ideas behind analyzing them, noting that several of those ideas have found use well beyond this one problem. The core claim driving the talk is a gap: mathematicians have a fairly complete picture of how to achieve expansion for simple groups, but almost no comparable understanding for groups that are not simple. Wigderson lays out concrete open questions in that territory and builds the group theory and representation theory background from scratch, assuming no prior exposure to the subject. Running just over two hours, the lecture moves from historical context into current research problems, aimed at a seminar audience but structured so a prepared newcomer can follow the whole arc.