Virtually planar groups (that is, those groups with a finite-index subgroup admitting a planar Cayley graph) exhibit many fairly unique coarse geometric properties. Often, we find that any one of these properties completely characterises this class of groups.
In this talk, I will survey some characterisations of virtually planar/virtual surface groups. Among other things, I will sketch why any group which is QI to a planar graph is necessarily virtually planar. Time permitting, I will discuss an upcoming new characterisation of virtual surface groups in terms of their coarse geometry.