The theory of C*-algebras developed in response to the advent of quantum theory in the early 1900’s, and have since developed into an interesting and active area of mathematics. When studying a physical system, we are interested in equilibrium states of the system – those which remain steady with the passage of time. In C*-algebraic models, such equilibrium states are called KMS-states, and are characterised by a simple-looking algebraic relation. This relation makes perfectly good sense even for C*-algebras that are not based on physical systems, and we are learning that the KMS states of a C*-algebra tend to capture very interesting structural information. I will discuss an excellent example of this phenomenon in the context of the graph C*-algebras that were introduced by Kumjian, Pask, Raeburn and Renault in the late 1990’s and have since become a major research industry worldwide. This is joint work with Astrid an Huef, Marcelo Laca and Iain Raeburn.