We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, V . In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of H V = - Δ + V and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution e - i H V t ψ 0 , for data ψ 0 , which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion of the effective large time evolution for the nonlinear Schrödinger - Gross Pitaevskii equation for small amplitude initial conditions, ψ 0 . The effective dynamics are governed by a nonlinear Dirac system.