Green functions have always played a prominent role in many-body physics. In particular one and two-body Green's function deliver a wealth of information about a physical system, such as ground-state energies, ionization potentials, electron affinities, spectral functions, excitation energies, absorption spectra, etc. Therefore the development of approximate methods to calculate the one-body Green's function has been an active research topic in many-body physics since the 60's, and many routes have been explored in order to find increasingly accurate Green's functions. A very popular class of methods is based on the solution of an integral equation for the one-body Green's function containing an effective potential, the so-called self-energy, which needs to be approximated. The well-known GW approximation belongs to this class; this approximation is the method of choice for calculating band structures, but it also shows several shortcomings, such as the wrong description of satellites in photoemission spectra, in particular in so-called strongly correlated materials. Therefore more refined levels of approximations are needed to keep the pace with the advances made in experiment. Recently much progress has been made in this direction both by going beyond standard methods and also exploring completely novel routes to calculate Green's functions. A new wave of original ideas, understanding, and solutions, has pervaded the field in these last years. Many new developments have occurred since the last successful "Green's function methods" workshop we held in Lausanne in May 2019. In particular the community has seen:

1) Many novel approaches to describe physical observables from Green’s functions such as correlation satellites in optical spectra [1], exciton band structures [2], inelastic x-ray scattering [3] and photoemission from the 3-body Green’s function
2) Many efforts to combine many-body perturbation theory with other theories such as dynamical mean-field theory [4,5,6] and parquet theory [7] and to include new physics such as electron-phonon coupling [8,9] and non-collinearity [10].
3) The development of vertex corrections both in the self-energy [11,12,13] and in the screening [14,15]

Therefore it is timely to gather these novel explorations in a fifth edition of this workshop, which previous editions have been very successful. In particular we would like to answer the following questions.

1) For which observables and/or materials are vertex corrections important?
2) What can the "ab initio community" learn from the "DMFT community" and vice versa?
3) How are the self-consistency issues dealt with in different approaches?
4) What are the best strategies to go beyond perturbation theory (in the density, the screening, the potential)?