We will give a gentle introduction to the world of big mapping class groups. These arise as groups of topological symmetries of: Infinite-type surfaces (dimension 2), locally finite, infinite graphs (dimension 1), and closed subsets of Cantor sets, i.e. second countable Stone spaces (dimension 0). These groups are all uncountable and come equipped with a topology that makes them into non-locally compact (aka "huge") Polish groups. We will survey some of the rich and interesting interplay between the topological, geometric, and algebraic aspects of these groups. Our main motivation will be to propose these groups as a fertile testing ground for extending the tools of geometric group theory beyond finitely (or compactly) generated groups to general "huge" groups; however, we will also mention potential connections to other fields of geometry, topology, and dynamics.