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Self-consistent diagrammatic approximations to the Anderson or Kondo impuritymodel, using an exact pseudoparticle representation of the impurity states, arereviewed. We first discuss the infrared exponents of the pseudoparticlepropagators as indicators of Fermi liquid behavior through their dependence onthe impurity occupation and on magnetic field. Then we discuss the Non-CrossingApproximation (NCA), identifying its strengths, but also its fundamentalshortcomings. Physical arguments as well as a perturbative renormalizationgroup analysis suggest that an infinite parquet-type resummation oftwo-particle vertex diagrams, the Conserving T-Matrix Approximation (CTMA) willcure the deficiencies of NCA. We review results on the pseudoparticle spectralfunctions, the spin susceptibility and the impurity electron spectral function,supporting that the CTMA provides qualitatively correct results, both in thehigh-temperature regime and in the strong coupling Fermi liquid regime at lowtemperatures.Comment: 10 pages, invited article, to appear in a special edition of JPSJ  "Kondo Effect - 40 Years after the Discovery", published version, some minor  typos correcte

Conserving Diagrammatic Approximations for Quantum Impurity Models: NCA and CTMA