  We present selected results from combined analytical and numerical studies of
the Anderson-impurity model within the framework of infinite order perturbation
theory with respect to the hybridization. Our approximation goes considerably
beyond the well known non\-cross\-ing approximation (NCA): The re-summations
include skeleton diagrams up to infinite order of the cross\-ing variety,
including all $1/N^2$ contributions ($N =$ degeneracy of the magnetic state).
We demonstrate, in a comparison with NCA theory, changes in the threshold
behaviour of local propagators, a qualitative improvement of spectral
properties and a clear progress in attaining local Fermi-liquid properties.
Further applications of the post-NCA theory are pointed out.
