In this paper I give a detailed account of an ab initio methodology for describing strong electronic correlations in nanoscale devices hosting transition metal atoms with open d- or f-shells. The method combines Kohn-Sham density functional theory for treating the weakly interacting electrons on a static mean-field level with non-perturbative many-body methods for the strongly interacting electrons in the open d- and f-shells. An effective description of the strongly interacting electrons in terms of a multi-orbital Anderson impurity model is obtained by projection onto the strongly correlated subspace properly taking into account the non-orthogonality of the atomic basis set. A special focus lies on the ab initio calculation of the effective screened interaction matrix U for the Anderson model. Solution of the effective Anderson model with the one-crossing approximation or other impurity solver techniques yields the dynamic correlations within the strongly correlated subspace giving rise e.g. to the Kondo effect. As an example the method is applied to the case of a Co adatom on the Cu(0 0 1) surface. The calculated low-bias tunnel spectra show Fano-Kondo lineshapes similar to those measured in experiments. The exact shape of the Fano-Kondo feature as well as its width depend quite strongly on the filling of the Co 3d-shell. Although this somewhat hampers accurate quantitative predictions regarding lineshapes and Kondo temperatures, the overall physical situation can be predicted quite reliably.