The moduli space \(S_{g, r}\) of \(r\)-spin curves parametrize \(r\)-th order roots of the canonical bundles of curves of genus \(g\). This space is an interesting cover of the moduli space of curves. For instance it carries a highly non-trivial virtual fundmental class whose numerical properties lead to a well-known prediction of Witten. I will discuss various topics related to the birational geometry and intersection theory of these spaces, focusing both on the more classical case of theta-characteristics (\(r=2\)), as well as on the higher order analogues.