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 the resulting material is characterised using a range of surface-sensitive techniques as well as extensive analysis of electronic transport phenomena. Comparisons made between the graphene produced by the initial and improved processes indicate an order of magnitude increase in structural coherence as a result of the changes made, with associated improvements in electronic characteristics. Magnetotransport measurements are presented which demonstrate the two-dimensional nature of the material. These can also be used to extract values for rates associated with the various scattering mechanisms present, which in turn give insight into the electronic coherence, linking this with the physical properties. Consideration of electronelectron interaction effects is required in order to fully explain the magnetotransport behaviour seen. Detailed consideration of surface properties seen at intermediate stages in the decomposition process (including a previously unreported surface reconstruction) is used to suggest a change to established growth procedures which if successful has the potential to further improve the quality of material.