Rhombohedral α-Fe₂O₃ has been studied by using density-functional theory (DFT) and the generalized gradient approximation (GGA). For the chosen supercell all possible magnetic configurations have been taken into account. We find an antiferromagnetic ground state at the experimental volume. This state is 388 meV/(Fe atom) below the ferromagnetic solution. For the magnetic moments of the iron atoms we obtain 3.4μB, which is about 1.5μB below the experimentally observed value. The insulating nature of α-Fe₂O₃ is reproduced, with a band gap of 0.32 eV, compared to an experimental value of about 2.0 eV. Analysis of the density of states confirms the strong hybridization between Fe 3d and O 2p states in α-Fe₂O₃. When we consider lower volumes, we observe a transition to a metallic, ferromagnetic low-spin phase, together with a structural transition at a pressure of 14 GPa, which is not seen in experiment. In order to take into account the strong on-site Coulomb interaction U present in Fe₂O₃ we also performed DFT+U calculations. We find that with increasing U the size of the band gap and the magnetic moments increase, while other quantities such as equilibrium volume and Fe-Fe distances do not show a monotonic behavior. The transition observed in the GGA calculations is shifted to higher pressures and eventually vanishes for high values of U. Best overall agreement, also with respect to experimental photoemission and inverse photoemission spectra of hematite, is achieved for U=4eV. The strength of the on-site interactions is sufficient to change the character of the gap from d-d to O-p-Fe-d.