  While key effects of the many-body problem---such as Kondo and Mott
physics---can be understood in terms of on-site correlations, non-local
fluctuations of charge, spin, and pairing amplitudes are at the heart of the
most fascinating and unresolved phenomena in condensed matter physics. Here, we
review recent progress in diagrammatic extensions to dynamical mean-field
theory for ab initio materials calculations. We first recapitulate the quantum
field theoretical background behind the two-particle vertex. Next we discuss
latest algorithmic advances in quantum Monte Carlo simulations for calculating
such two-particle quantities using worm sampling and vertex asymptotics, before
giving an introduction to the ab initio dynamical vertex approximation
(AbinitioD$\Gamma$A). Finally, we highlight the potential of AbinitioD$\Gamma$A
by detailing results for the prototypical correlated metal SrVO$_3$.
