The charmonium system is composed of a charm quark and an anti-charm quark bound together by the strong force into various configurations. It is arguably the simplest system in which to study the strong force. It is analogous to hydrogen or positronium in the study of the electromagnetic force. But over the last decade a number of new phenomena have begun to appear in the charmonium system that seem to point beyond this simple picture. I will review a few of these phenomena, referred to as the XYZ states, and I will discuss how the BESIII Experiment, based in Beijing, China, is working to understand these puzzles.