The canonical cover of an Enriques or a Coble surface in characteristic two, when it is inseparable and the surface has only rational double points, is birationally isomorphic to a supersingular K3 surface. In this chapter, after introducing some basic facts from the theory of supersingular K3 surfaces, we will study some constructions of Enriques and Coble surfaces with interesting groups of automorphisms as quotients of a supersingular K3 surface by a rational vector field.