Notwithstanding the graphitization of SiC under high thermal treatment can take place for all SiC surfaces, the quality of the resulting graphene as well as its structural and electrical characteristics strongly depend on the SiC face where growth has taken place. In this paper we use the density functional theory to analyze the structural and electronic properties of epitaxial graphene grown on three different SiC planes. Calculations are presented for the (6√ 3× 6√ 3) R30-reconstructed SiC (0001) surface (Si face) as well as the nonpolar SiC (11-20) and SiC (1-100) planes. We argue that the formation of a strongly-bound interface buffer layer is an exclusive property of the SiC (0001) surface. Moreover, our results indicate that nonpolar planes give rise to graphene with a nearly ideal low-energy spectrum.