The mapping class groups of infinite-type surfaces are examples of automorphism groups of countable structures, coming from the topology of the underlying surface. Questions on their structure and large-scale geometry have recently caught the interest of a number of low-dimensional topologists and geometric group theorists, who have begun to draw on tools from descriptive set theory to answer them. This talk will be both an introduction to these groups for the non-topologist, and a pitch of some unexplored research directions, with an aim to draw connections between these groups and other large transformation groups.