The nature of the bonding in the binary transition-metal carbonyl complex has been analyzed by topological approaches (atoms in molecules (AIM) and electron localization function (ELF)) from a series of calculations carried out at the hybrid Hartree-Fock/DFT level (B3LYP). It is shown that the interaction between a transition metal and CO should be characterized as a dative bond, in which the monosynaptic basin of the carbon plays the role of the disynaptic basin connecting the metal core to the carbon atom. For all atoms except Cr, Mn, and Cu, the multiplicity of the ground state is given by applying Hund's rule to the maximal core occupancy (i.e., [Ar]Cn+2): high-spin complexes for n < 4, low-spin for n > 5, spin-conserved for n = 4, 5, 9. The charge transfers and the spin density on the ligand are rationalized by resonance structures of the same multiplicity. In all complexes except CrCO and CuCO, the ELF function in the core has a local cylindrical symmetry that in turn favors a linear structure; moreover, 2 electrons are available for the charge transfer toward the CO moiety and for the metal nonbonding valence basin. In CrCO and CuCO whose cores have a spherical symmetry, only one electron can be shared by the net transfer and the nonbonding valence basin. The maximization of the charge transfer implies a bent geometry. Finally, we propose two new donation-back-donation schemes based on the AIM and ELF partitions. In the ELF framework, the net charge transfer is almost equal to the pi back-donation, the sigma-donation being negligible.