Ground-state properties of materials, such as the atomic structure, the total energy, the electronic density and the phonon frequencies, can be investigated from ab initio with high accuracy by density-functional theory (DFT). On the other hand, DFT cannot describe excited-state properties such as the bandgap, the bandplot or the excitations sampled in ARPES spectroscopy. One needs to go beyond. A possibility is represented by Hedin's GW approximation in the framework of many-body quantum field theory. We will present ab initio GW calculations of the electronic structure, the momentum distribution and the conductance. The presented systems will range from 3D standard Fermi liquid metals (Na) and more exotic Mott or excitonic insulators (VO2, TiSe2), to low dimensional systems, such as 2D graphene, 1D nanowires and 0D organic molecules. Our results will be compared to ARPES, IXSS, STS and quantum transport experiments.