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Historically, the GW approach was put forward by Hedin as the simplestapproximation to the so-called Hedin equations. In Section 2, we will derivethese Hedin equations from a Feynman-diagrammatical point of view. Section 3.1shows how GW arises as an approximation to the Hedin equations. In Section 3.2,we briefly present some typical GW results for materials, includingquasiparticle renormalizations, lifetimes, and band gap enhancements. InSection 4, the combination of GW and DMFT is summarized. Finally, as aprospective outlook, ab initio dynamical vertex approximation D$\Gamma$A isintroduced in Section 5 as a unifying scheme for all that: GW, DMFT andnon-local vertex correlations beyond.

Hedin Equations, GW, GW+DMFT, and All That