This thesis investigates mapping class groups of infinite-type surfaces, also called big mapping class groups, by studying their actions on certain graphs whose vertices are arcs and curves on the underlying surface. In particular, we show that the extended mapping class group of any surface with a finite, positive number of punctures is isomorphic to the relative arc graph of that surface; that the mapping class group of any translatable surface is quasi-isometric to that surface's translatable curve graph; and that the mapping class group of a sphere minus a Cantor set is quasi-isometric to that surface's loop graph.