All of this remarkable progress rests on the foundation of equilibrium statistical mechanics. Phases and phase transitions are only sharply defined for macroscopic systems in the thermodynamic limit, and statistical mechanics allows us to make useful predictions about such macroscopic systems with many (~ 10) constituent particles. A fundamental assumption of statistical mechanics is that systems generically reach a state of thermal equilibrium (such as the Gibbs state) which can be characterized by only a few parameters such as temperature or a chemical potential. Traditionally, phase structure is studied by examining the behavior of ``order parameters" in equilibrium states. At zero temperature, these are evaluated in the ground state of the system, and different phases correspond to different quantum orders (topological or otherwise). Thermal equilibrium strongly constrains the allowed orders at finite temperatures. In general, thermal fluctuations at finite temperatures reduce the long-ranged quantum correlations present in ordered phases and, in lower dimensions, can destroy order altogether. As an example, the Peierls-Mermin-Wagner theorems prove that a one dimensional system cannot spontaneously break a continuous symmetry at any non-zero temperature.