Epitaxial graphene growth on SiC is investigated using low-energy electron microscopy (LEEM) and first-principles calculations. LEEM is one of the most powerful tools to identify the thickness of graphene on SiC with a good spatial resolution. With the help of such LEEM, the thickness-dependent physical properties are identified by various experiments. It is shown that epitaxial graphene sheets continue even over steps of the substrate, and that a new graphene sheet often grows from step edges while the surface morphology changes drastically. Furthermore, the first-principles calculations also show the energetics of the epitaxial graphene growth on SiC. It is expected that the fine control of epitaxial graphene growth on SiC will open the way to novel graphene devices in the post-scaling era of the ultra-large-scale integrations (ULSI).