Graphene is a single atomic layer of sp.sup.2-bonded C atoms densely packed into a two-dimensional honeycomb crystal lattice. A method of forming structurally perfect and defect-free graphene films comprising individual mono crystalline domains with in-plane lateral dimensions of up to 200 .mu.m or more is presented. This is accomplished by controlling the temperature-dependent solubility of interstitial C of a transition metal substrate having a suitable surface structure. At elevated temperatures, C is incorporated into the bulk at higher concentrations. As the substrate is cooled, a lowering of the interstitial C solubility drives a significant amount of C atoms to the surface where graphene islands nucleate and gradually increase in size with continued cooling. Ru(0001) is selected as a model system and electron microscopy is used to observe graphene growth during cooling from elevated temperatures. With controlled cooling, large arrays of macroscopic single-crystalline graphene domains covering the entire transition metal surface are produced. As the graphene domains coalesce to a complete layer, a second graphene layer is formed, etc. By controlling the interstitial C concentration and the cooling rate, graphene layers with thickness up to 10 atomic layers or more are formed in a controlled, layer-by-layer fashion.