  We study the existence of edge modes in gapped Moir\'e superlattices in
graphene monolayer ribbons. We find that the superlattice bands acquire finite
Chern numbers, which lead to a Valley Hall Effect. The presence of dispersive
edge modes is confirmed by calculations of the band structure of realistic
nanoribbons using tight binding methods. These edge states are only weakly
sensitive to disorder, as short-range scattering processes lead to mean free
paths of the order of microns. The results explain the existence of edge
currents when the chemical potential lies within the bulk superlattice gap, and
offer an explanation for existing non-local resistivity measurements in
graphene ribbons on boron nitride.
