The mapping torus of a group is an important object in geometric group theory that exhibits a wide range of behaviour. After giving a brief background to this topic, I will talk about recent joint work with François Dahmani, where we showed that for an automorphism of a free product that is “hyperbolic” relative to the free product structure, the mapping torus acts relatively geometrically on a hyperbolic CAT(0) cube complex. This is a generalisation of an analogous result by Mark Hagen and Dani Wise for mapping tori of free groups.