With the decrease of the dimensions of electronic devices, the role played by electrical contacts is ever increasing, eventually coming to dominate the overall device volume and total resistance. This is especially problematic for monolayers of semiconducting transition metal dichalcogenides (TMDs), which are promising candidates for atomically thin electronics. Ideal electrical contacts to them would require the use of similarly thin electrode materials while maintaining low contact resistances. Here we report a scalable method to fabricate ohmic graphene edge contacts to two representative monolayer TMDs, MoS2 and WS2. The graphene and TMD layer are laterally connected with wafer-scale homogeneity, no observable overlap or gap, and a low average contact resistance of 30 k Omega.mu m. The resulting graphene edge contacts show linear current voltage (I-V) characteristics at room temperature, with ohmic behavior maintained down to liquid helium temperatures.