Translation surfaces arise naturally in the study of several dynamical and algebro-geometric problems. They can often be built simply from polygons by gluing their edges via isometries. This elementary perspective makes it easy to associate to a translation surface topological or combinatorial data that are invariant under affine deformations of the surface. The most basic such invariants are the number and type of cone angle singularities present on the surface. In this talk I will discuss two related invariants. Given a compact surface, there is a hyperbolic tessellation whose tiles are associated to canonical triangulations of the surface, which have the cone points as vertices. For non-compact surfaces, more complicated singular behavior can arise, which is captured by a Hausdorff topological space that generalizes the unit tangent bundle of the surface. I will present work from my Ph.D. thesis and joint work with F. Valdez.