Graphene and carbon nanotube (CNT) structures have promise for many electronic device applications and both have been grown on SiC through the decomposition of the substrate. It is well known that both graphene and aligned CNTs are grown under similar conditions with overlapping temperature and pressure ranges, but a fundamental understanding of the two types of growths is actively being researched. Moreover, various technical challenges need to be overcome to achieve improvement in the electronic and structural quality of these carbon-based nanostructures on SiC. Specifically, an understanding and control of the SiC surface graphitization process and interface structure needs to be established. In this review, we focus on graphene growth on SiC (0 0 0 1) (Si-face) as a model system in comparison with aligned CNT growth on SiC. The experimental growth aspects for graphene growth, including vacuum and ambient growth environments, and growth temperature are summarized, then proposed decomposition and growth mechanisms are discussed. Both thermal and chemical decomposition processes are presented and special emphasis is given to the role of growth process variations between laboratories. The chemical reactions driving the graphitization process and ultimately the carbon nanostructure growth on SiC are discussed. It is suggested that the composition of the residual gases in the growth environment is a critical parameter and that gas composition at the growth temperature should be monitored.