The electronic structures and the chemical-bonding mechanism of Cu3N, Cu3NPd and related Cu(I) compounds such as Cu2O, are studied on the basis of band-structure calculations, using both the linearized augmented plane wave and linear combination of atomic orbitals (LCAO) methods. In accordance with experimental observations, Cu3N is found to be a semiconductor, while Cu3NPd should exhibit a semimetallic conductivity. The chemical bonding is investigated using various methods, among them are the valence charge partitioning scheme of Bader and a basis set reduction technique built on the LCAO method. A partly ionic, partly covalent bonding is found. The admixture of the Cu (4s, 4p) states to the Cu 3d-N 2p bands resulted to be essential for the covalent bonding effect, since pure 3d-2p bands, with bonding and antibonding states fully, occupied, do not lead to a covalent energy gain. This specific hybridization appears to be the origin of the twofold dumbbell like Cu(I) coordination observed in Cu3N and other Cu(I) compounds In Cu3NPd, a covalent to metallic bonding between the Cu3N host crystal and the interstitial Pd atoms is found, which is mainly caused by Pd 5s and 5p states hybridizing Cu 3d states.