Though the term translation surface is relatively new, the notion is quite old (Abelian differential) and in fact predates the notion of hyperbolic surface. The term translation surface emphasizes the (G, X) point of view introduced to their study by Thurston in about 1980. Since then, ideas from both topology and dynamics have revolutionized the study of these structures. In this talk I will describe some of these ideas with preference given to those ideas that connect to both Delaunay triangulations and hyperbolic geometry. This talk is based on joint work with Allen Broughton. See https://arxiv.org/abs/1003.1672[-]