For the treatment of interacting electrons in crystal lattices, approximations based on thepicture of effective sites, coupled in a self-consistent fashion, have proven very useful.Particularly in the presence of strong local correlations, a local approach to theproblem, combining a powerful method for the short-ranged interactions with thelattice propagation part of the dynamics, determines the quality of results to alarge extent. For a considerable time the noncrossing approximation (NCA) indirect perturbation theory, an approach originally developed by Keiter for theAnderson impurity model, was a standard for the description of the local dynamics ofinteracting electrons. In the last couple of years exact methods like the numericalrenormalization group (NRG), as pioneered by Wilson, have surpassed this approximationas regarding the description of the low-energy regime. We present an improvedapproximation level of direct perturbation theory for finite Coulomb repulsionU, the crossing approximation 1 (CA1), and discuss its connections with other generalizations ofNCA. CA1 incorporates all processes up to fourth order in the hybridization strengthV in a self-consistent skeleton expansion, retaining the full energy dependence of the vertexfunctions. We reconstruct the local approach to the lattice problem from the point of viewof cumulant perturbation theory in a very general way and discuss the proper use ofimpurity solvers for this purpose. Their reliability can be tested in applications to, forexample, the Hubbard model and the Anderson-lattice model. We point out shortcomingsof existing impurity solvers and improvements gained with CA1 in this context.