van der Waals epitaxial growth of graphene on c-plane (0001) sapphire by CVD without a metal catalyst is presented. The effects of CH(4) partial pressure, growth temperature, and H(2)/CH(4) ratio were investigated and growth conditions optimized. The formation of monolayer graphene was shown by Raman spectroscopy, optical transmission, grazing incidence X-ray diffraction (GIXRD), and low voltage transmission electron microscopy (LVTEM). Electrical analysis revealed that a room temperature Hall mobility above 2000 cm(2)/V·s was achieved, and the mobility and carrier type were correlated to growth conditions. Both GIXRD and LVTEM studies confirm a dominant crystal orientation (principally graphene [10-10] || sapphire [11-20]) for about 80-90% of the material concomitant with epitaxial growth. The initial phase of the nucleation and the lateral growth from the nucleation seeds were observed using atomic force microscopy. The initial nuclei density was ~24 μm(-2), and a lateral growth rate of ~82 nm/min was determined. Density functional theory calculations reveal that the binding between graphene and sapphire is dominated by weak dispersion interactions and indicate that the epitaxial relation as observed by GIXRD is due to preferential binding of small molecules on sapphire during early stages of graphene formation.