We study the electronic structure of graphene with a single substitutional vacancy using a combination of the density-functional, tight-binding and impurity Green's function approaches. Density-functional studies are performed with the all-electron spin-polarized linear augmented plane wave (LAPW) method. The three sp 2σ dangling bonds adjacent to the vacancy introduce localized states (Vσ) in the mid-gap region, which split due to the crystal field and a Jahn-Teller distortion, while the p zπ states introduce a sharp resonance state (Vπ) in the band structure. For a planar structure, symmetry strictly forbids hybridization between the σ and the π states, so that these bands are clearly identifiable in the calculated band structure. As to the magnetic moment of the vacancy, the Hund's rule coupling aligns the spins of the four localized Vσ 1 Vσ 2 and Vπ electrons, resulting in an S = 1 state, with a magnetic moment of 2μB, which is reduced by about 0.3μB due to the anti-ferromagnetic spin polarization of the π band itinerant states in the vicinity of the vacancy. This results in the net magnetic moment of 1.7μB. Using the Lippmann-Schwinger equation, we reproduce the well-known ̃1/r decay of the localized Vπ wave function with distance, and in addition, find aninterference term coming from the two Dirac points, previously unnoticed in the literature. The long-range nature of the Vπ wave function is a unique feature of the graphene vacancy and we suggest that this may be one of the reasons for the widely varying relaxed structures and magnetic moments reported from the supercell band calculations in the literature. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.