The space of subgroups Sub(G) of a countable group G is a compact space on which the group acts by homeomorphisms. It encodes many aspects of a group's behaviour, often providing a way of detecting or ruling out actions of G. I will give an overview of what is known about the topological and dynamical structure of Sub(G) for various classes of groups, focusing on the cases where G exhibits notions of non-positive curvature. In particular, I will discuss some recent work in which we show that, when G is cubulated, virtually compact special, non-elementary and irreducible, the action of G on the closure of its infinite index convex cocompact subgroups is chaotic. This talk is based on work in progress with Damien Gaboriau, Mark Hagen and Zach Munro.