Employing density-functional calculations we study single and double graphene layers on Si-and C-terminated 1 × 1 -6H-SiC surfaces. We show that, in contrast to earlier assumptions, the first carbon layer is covalently bonded to the substrate, and cannot be responsible for the graphenetype electronic spectrum observed experimentally. The characteristic spectrum of free-standing graphene appears with the second carbon layer, which exhibits a weak van der Waals bonding to the underlying structure. For Si-terminated substrate, the interface is metallic, whereas on C-face it is semiconducting or semimetallic for single or double graphene coverage, respectively.PACS numbers: 68.35. Ct, 68.47.Fg, The last years have witnessed an explosion of interest in the prospect of graphene-based nanometer-scale electronics [1,2,3,4]. Graphene, a single hexagonally ordered layer of carbon atoms, has a unique electronic band structure with the conic "Dirac points" at two inequivalent corners of the two-dimensional Brillouin zone. The electron mobility may be very high and lateral patterning with standard lithography methods allows device fabrication [1]. Two ways of obtaining graphene samples have been used up to now. In the first "mechanical" method, the carbon monolayers are mechanically split off the bulk graphite crystals and deposited onto a SiO 2 /Si substrate [4]. This way an almost "freestanding" graphene is produced, since the carbon monolayer is practically not coupled to the substrate. The second method uses epitaxial growth of graphite on singlecrystal silicon carbide (SiC). The ultrathin graphite layer is formed by vacuum graphitization due to Si depletion of the SiC surface [5]. This method has apparent technological advantages over the "mechanical" method, however it does not guarantee that an ultrathin graphite (or graphene) layer is electronically isolated from the substrate. Moreover, one expects a covalent coupling between both which may strongly modify the electronic properties of the graphene overlayer. Yet, experiments show that the transport properties of the interface are dominated by a single epitaxial graphene layer [1,2]. Most surprisingly, the electronic spectrum seems not to be affected much by the substrate. As in free-standing graphene one observes the "Dirac points" with the linear dispersion relation around them. The electron dynamics is governed by a Dirac-Weyl Hamiltonian with the Fermi velocity of graphene replacing the speed of light. This leads to an unusual sequence of Landau levels in a magnetic field and hence to peculiar features in the quantum Hall effect [1,4].