Big mapping class groups refer to the mapping class groups of infinite-type surfaces, i.e. those surfaces whose fundamental group is not finitely generated. The adjective “big” refers to the underlying surface, but big mapping class groups are also bigger than their finite-type counterparts in other ways; for instance, they are uncountable groups, and they are also non-locally compact as topological groups. Despite this, and somewhat surprisingly, these groups can be geometrically small. We will discuss several ways in which uncountable groups can be small, and go over recent results placing various big mapping class groups into these categories. Part of the work discussed is joint with Justin Lanier.