The electronic structure of covellite (CuS) is analyzed on the basis of density functional theory calculations. The nature of holes in the valence band, as well as the so far much debated question of the appropriate oxidation formalism for this conductor, is discussed. The role of S-S bonds and the anomalous coordination of one type of sulfur atom (hypervalency) are considered. It is suggested that the low-temperature transition is mostly a symmetry-lowering process slightly stabilizing the Cu-S network.