Dynamical Vertex Approximation

We give an elementary introduction to a recent diagrammatic extension ofdynamical mean field theory (DMFT) coined dynamical vertex approximation(D$\Gamma$A). This approach contains the important local correlations of DMFT,giving, among others, rise to quasiparticle renormalizations, Mott-Hubbardtransitions and magnetism, but also non-local correlations beyond. The latterare at the very essence of many physical phenomena in strongly correlatedelecectron systems. As correlations are treated equally on all length scales,D$\Gamma$A allows us to describe physical phenomena such as magnons, quantumcriticality, and the interplay between antiferromagnetism andsuperconductivity. We review results hitherto obtained for the Hubbard model indimensions d=3, 2, and 1.