Rigidity theorems in geometric group theory prove that a group’s geometric type determines its algebraic type, typically up to virtual isomorphism. We study graphically discrete groups, which impose a discreteness criterion on the automorphism groups of graphs the group acts on and are well suited to studying rigidity problems. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds; nonabelian free groups are non-examples. We will present new families of graphically discrete groups and demonstrate this property is not a quasi-isometry invariant. We will discuss rigidity phenomena for free products of graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.