We report on the properties of the (0001) covellites surfaces, which we investigate by periodic slab density functional theory calculations. The absolute surface energies have been computed for all bulk terminations, showing that surfaces terminated by the flat CuS layer are associated with the lowest surface energy. Cleavage is predicted to occur across the [0001] interlayer Cu-S bond. The surfaces obtained by lowest energy cleavage are analyzed in terms of the atomic vertical relaxation, workfunction, and surface band structure. Our study predicts the presence of a shallow pz-derived surface state located 0.26 eV below the Fermi level, which is set to play an important role in the surface reactivity of covellite.