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 Toulouse in June 2017. In particular the community has seen:

1) An explosive growth of papers in which Green's functions are used within quantum chemistry. [1-3]

2) Many efforts to combine Green's functions with other theories such as density-functional theory, reduced-density-matrix functional theory, quantum Monte Carlo, density-matrix renormalization group, etc. [4-8]

Therefore it is timely to gather these novel explorations in a workshop.

In particular we would like to answer the following questions.

1) What are the issues/problems of the cohabitation of GF and wave-function based approaches?

2) Which knowledge obtained from the one-body GF can be exported to the calculation of higher-order GF calculations?

3) What can the "solid state physics community" learn from the "quantum chemistry community" and vice versa?

4) How the self-consistency issues are dealt with in different approaches?

5) What are the best strategies to go beyond perturbation theory (in the density, the screening, the potential)?