Abstract: We discuss continuum Schroedinger operators which are basic models of 2D-materials, like graphene; in its bulk form or deformed by edges (sharp terminations or domain walls). For non-magnetic and strongly non-magnetic systems we discuss the relationship to effective tight binding (discrete) Hamiltonians through a result on strong resolvent convergence. An application of this convergence is a result on the equality of topological (Fredholm) indices associated with continuum and discrete models (for bulk and edge systems). Finally, we discuss the construction of edge states in continuum systems with domain walls. Away from the tight binding regime there are resonant phenomena, and we conjecture that there are meta-stable (finite lifetime, but long-lived) edge states which slowly diffract into the bulk.