Given a surface S, we consider two associated graphs: the by now classical curve graph, on which the mapping class group of S acts, and the recently defined fine curve graph, on which the homeomorphism group of S acts. In both cases, for a group element, having positive asymptotic translation length (i.e. displacing every vertex at linear speed, roughly speaking) corresponds to having interesting topological/dynamical properties. I will discuss joint work with Sebastian Hensel and Frédéric Le Roux where we establish relations and study differences between the asymptotic translation lengths of homeomorphisms and mapping classes