We directly measure the electronic structure of twisted graphene/MoS2 van der Waals heterostructures, in which both graphene and MoS2 are monolayers. We use cathode lens microscopy and microprobe angle-resolved photoemission spectroscopy measurements to image the surface, determine twist angle, and map the electronic structure of these artificial heterostructures. For monolayer graphene on monolayer MoS2, the resulting band structure reveals the absence of hybridization between the graphene and MoS2 electronic states. Further, the graphene-derived electronic structure in the heterostructures remains essentially intact, irrespective of the twist angle between the two materials. In contrast, however, the electronic structure associated with the MoS2 layer is found to be twist-angle dependent; in particular, the relative difference in the energy of the valence band maximum at 􏰀 ̄ and K ̄ of the MoS2 layer varies from approximately 0 to 0.2 eV. Our results suggest that monolayer MoS2 within the heterostructure becomes predominantly an indirect band-gap system for all twist angles except in the proximity of 30◦. This result enables potential band-gap engineering in van der Waals heterostructures comprised of monolayer structures.