The molecular structure of the different isomers arising from the interaction of Cu and both MgO and CaO molecules has been determined from all-electron ab initio self-consistent field, SCF, and complete active space self-consistent field, CASSCF, wave functions. Only linear stable structures are found as characterized through the vibrational analysis. The dissociation energy with respect to the isolated molecules was obtained by explicitly considering the electronic correlation effects through second-order perturbation theory, MP2, or multireference configuration interaction calculations. The role of electronic correlation is very important due to the near degeneracies existing in the isolated MgO and CaO molecules where the SCF description is not adequate enough. A set of theoretical techniques has been used to analyze the chemical bond in these molecular complexes. All the results indicate that the bonding in the O- and M-bonded complexes is very different. This fact may be interpreted from the different behavior of the dipole moment curve and from simple arguments such as the importance of different valence bond resonating structures.