The formation energy, geometry, and electronic structure of isolated oxygen and aluminum vacancies in bulk and on the (0001) surface of corundum ( α − Al 2 O 3 ) have been investigated by means of periodic calculations in the framework of density functional theory within the generalized gradient approximation and large supercells. The energy cost to form an oxygen vacancy in the bulk is estimated to be of the order of 10 eV, whereas that corresponding to the formation of Al vacancies is found to be at least a 30% larger. The relaxation of the material is rather small for both defects. The removal of an oxygen atom in bulk α − Al 2 O 3 is accompanied by the appearance of an impurity level in the gap, which is a strong indication of electron localization. This has been further confirmed by integration of the density of states in the energy interval corresponding to the impurity level and by several other theoretical analyses. For the α − Al 2 O 3 ( 0001 ) surface, the formation of oxygen and aluminum vacancies exhibits many similarities with the bulk; the energy cost to form Al vacancies is much larger than for O vacancies and, in both cases, it is accompanied by rather small atomic displacements. However, there are also significant differences between bulk and surface oxygen and aluminum vacancies. Thus the formation energy of one of these point defects in the surface is rather smaller as expected and, more importantly, the degree of electronic delocalization is also larger.