We introduce various graphs whose vertices correspond to (possibly marked) arc and curve systems on a compact surface, and their applications in the study of finite-type surface mapping class groups. Examples will include: the curve graph, the arc graph, the pants graph, and the marking graph. We will discuss the quasi-isometry types of such graphs, and a theorem classifying their (equivariant) quasi-isometry types by the set of subsurfaces intersecting every vertex. If time permits, we will also describe their hierarchically hyperbolic structure.