Edge states are solutions of energy conserving wave equations which are bounded and oscillatory (plane-wave like) in the direction of a line defect, and localized transverse to it. We discuss edge states in systems with honeycomb symmetry, which arise in two dimensional quantum materials such as graphene and their metamaterial analogues. We first consider the case of a “rational edge”, a line defect (domain wall or sharp termination) in the direction of a period lattice vector. We then discuss very recent work with P. Amenoagbadji on edge states which localize transverse to an irrational edge.