In this thesis single-crystal 4H silicon carbide is decomposed by annealing in ultra-high vacuum conditions, forming a surface layer consisting of a few atomic layers of graphitic carbon (graphene) arranged in known crystallographic register with the substrate material. The layers of graphene on the (0001) face form the subject of the thesis, and their surface morphology, crystallography and electronic transport properties are investigated in order to gain insight into the growth process with a view to improving the quality of the graphene (assessed in terms of lateral grain size, and coherence of electronic transport). Graphene quality is improved following changes to the annealing procedure based on understanding of growth mechanisms, and...