  A classical result in thermodynamic formalism is that for uniformly
hyperbolic systems, every H\"older continuous potential has a unique
equilibrium state. One proof of this fact is due to Rufus Bowen and uses the
fact that such systems satisfy expansivity and specification properties. In
these notes, we survey recent progress that uses generalizations of these
properties to extend Bowen's arguments beyond uniform hyperbolicity, including
applications to partially hyperbolic systems and geodesic flows beyond negative
curvature. We include a new criterion for uniqueness of equilibrium states for
partially hyperbolic systems with 1-dimensional center.

Dynamical Systems