Abstract: Translation surfaces are objects which arise naturally in several different mathematical domains, e.g. complex analysis, topology of surface homeomorphisms, and the study of rational polygonal billiards. The symmetries of a given translation surface M naturally give rise to a subgroup of SL(2,R) called the Veech group of M. For generic M the Veech group is trivial, however in many cases it is quite large. In particular M is called a lattice surface if its Veech group is a lattice in SL(2,R). In 1989 Veech constructed many examples of lattice surfaces and showed that they have very interesting dynamical properties. We will present recent joint work with John Smillie and Pascal Hubert extending some of Veech's results. No prior acquaintance with Veech surfaces will be assumed, and pictures will be shown.