Consider a mapping of the torus that stretches and compresses it in two directions. (These are called Anosov maps.) The lift of such a map to the universal cover is the action of a matrix in SL(2,Z) on the plane and the stretch factor is an eigenvalue of the matrix. Therefore only quadratic algebraic integers can be stretch factors of the torus.  For higher genus surfaces, the topology of the surface still imposes constraints on the possible algebraic degrees of the stretch factors, but now a wider variety of degrees may appear. In this talk, I will explain a construction that realizes stretch factors of all possible degrees.