The problem of the classical motion of a round object is typically presented using idealized setups in order to make it more tractable. Popular examples include a sphere moving on a perfectly flat inclined plane. Here, we focus on a rolling object and show that more realistic cases of curved surfaces defined by a single variable and including friction are not only tractable, but also offer new physics. We show that the point at which the object may detach from the surface can be predicted accurately using simple methods. We check the accuracy of our theoretical calculations by performing experiments using tracks in the shape of four different conic curves. We observe very good agreement between the theoretical predictions and the experimental results. Our findings not only suggest that curved surfaces can be included in the presentation of motion to students, but that they also offer intriguing new scenarios for gaining physical insight.