For many dynamical systems, there are (at least) two geometrically interesting measures: the Lebesgue class measure, and the measure of maximal entropy, which is the limiting distribution of periodic orbits. Thermodynamic formalism provides a uniform way of studying these measures using the theory of equilibrium states. One particularly fruitful area of study is the thermodynamic formalism of geodesic flows in negative curvature, in which we know that these measures are the "most random" they can be. I will discuss what properties are known in this setting, and what we can show when we move beyond the setting of negative curvature to nonpositive curvature and CAT(0) spaces.