We report that the shape, orientation, edge geometry and thickness of chemical vapour deposition graphene domains can be controlled by the crystallographic orientations of Cu substrates. Under low-pressure conditions single layer graphene domains align with zig-zag edges parallel to a single <101> direction on Cu(111) and Cu(101), while bilayer domains align to two directions on Cu(001). Under atmospheric pressure conditions hexagonal domains also preferentially align. This discovery can be exploited to generate high-quality, tailored graphene with controlled domain thickness, orientations, edge geometries and grain boundaries.