  We review recent developments in electronic structure calculations that go
beyond state-of-the-art methods such as density functional theory (DFT) and
dynamical mean field theory (DMFT). Specifically, we discuss the following
methods: GW as implemented in the Vienna {\it ab initio} simulation package
(VASP) with the self energy on the imaginary frequency axis, GW+DMFT, and ab
initio dynamical vertex approximation (D$\Gamma$A). The latter includes the
physics of GW, DMFT and non-local correlations beyond, and allows for
calculating (quantum) critical exponents. We present results obtained by the
three methods with a focus on the benchmark material SrVO$_3$.
