I'll first give a broad introduction to big mapping class groups and why they may be of interest to set theorists. These groups are a rich class of non-archimedean Polish groups coming from the worlds of low-dimensional topology and geometric group theory. After this broad overview I will talk in more detail about some results (in parts joint with Mladen Bestvina, Ryan Dickmann, and Kasra Rafi) around automatic continuity properties of big mapping class groups of surfaces.