We discuss some recent progress on the theory of uniqueness of equilibrium states for geodesic flows. This talk will be suitable for a general mathematical audience, and will start with an intuitive overview of the classic results developed by luminaries such as Anosov, Bowen and Ruelle in the well understood setting of surfaces with negative curvature. I will then give an overview of some recent advances in the theory, including:
1) General machinery developed by Vaughn Climenhaga and myself, which gives "non-uniform" dynamical criteria for uniqueness of equilibrium states;
2) Joint work with Keith Burns, Vaughn Climenhaga and Todd Fisher, where we apply this machinery to geodesic flow on non-positive curvature surfaces;
3) Joint work with Jean-Francois Lafont and Dave Constantine, where we develop the theory of equilibrium states for geodesic flow on locally CAT(-1) spaces; these are geodesic metric spaces which generalize negative curvature Riemannian manifolds by having the "thin triangle" property.