  Self-consistent diagrammatic approximations to the Anderson or Kondo impurity
model, using an exact pseudoparticle representation of the impurity states, are
reviewed. We first discuss the infrared exponents of the pseudoparticle
propagators as indicators of Fermi liquid behavior through their dependence on
the impurity occupation and on magnetic field. Then we discuss the Non-Crossing
Approximation (NCA), identifying its strengths, but also its fundamental
shortcomings. Physical arguments as well as a perturbative renormalization
group analysis suggest that an infinite parquet-type resummation of
two-particle vertex diagrams, the Conserving T-Matrix Approximation (CTMA) will
cure the deficiencies of NCA. We review results on the pseudoparticle spectral
functions, the spin susceptibility and the impurity electron spectral function,
supporting that the CTMA provides qualitatively correct results, both in the
high-temperature regime and in the strong coupling Fermi liquid regime at low
temperatures.
