Let P be a rational polygon and consider a game of billiards played on P. A billiard ball bounces off the sides of P so that the angle of incidence equals the angle of reflection. If one wishes to study various dynamical aspects of this billiard system, e.g. periodic billiard orbits, then there is a useful “unfolding” construction that associates to the table P a flat surface S with cone points. Billiard orbits on P correspond to straightened, geodesic trajectories on S. The surfaces obtained via this construction are special examples of what are known as “translation surfaces”. In this talk, I will describe dynamical properties of the geodesic flow on special translation surfaces: the Veech examples and also certain non-Veech examples of Calta and McMullen in genus two. I will focus on results pertaining to closed trajectories on these surfaces and the elegant, often surprising ways that certain algebraic invariants can be used to study dynamical properties of the geodesic flow.