The past 15 years has seen an explosion of interest in the development of anisotropic Banach spaces on which the transfer operator associated to a hyperbolic dynamical system has good spectral properties. Starting with the work of Blank, Keller and Liverani for Anosov diffeomorphisms, this approach has since been broadened by a number of authors and applied to a wide variety of systems, including most recently, mathematical billiards. This talk will present the main ideas behind this analysis and discuss some recent applications of the method to proving the exponential decay of correlations for some billiard flows and the existence and properties of measures of maximal entropy for dispersing billiard maps.