For a surface with constant Gaussian curvature, various natural quantities grow exponentially quickly if curvature is negative, polynomially quickly if curvature is zero, and not at all if curvature is positive. I will describe this trichotomy for the number of (homotopy classes of) closed geodesics; then I will explain how Margulis proved even more precise asymptotic estimates in the case of negative curvature, and how his results have been extended to examples where curvature is zero or even positive in some places. Finally, I will describe joint work with Gerhard Knieper and Khadim War in which we obtain Margulis asymptotics for surfaces without conjugate points.