To calculate the bands including electron-electron interaction many-body effects, one can resort to so-called Greens function methods. Indeed, knowledge of the Greens function of a system provides both ground (the total energy) and also excited state observables of the system. The poles of the Greens function are the quasiparticle energies, the bands of a solid. The Greens function can be calculated by solving the Dyson equation once the self-energy of the system is known. For real systems like solids, the self-energy is a very complex quantity and usually approximations are needed to solve the problem. One such approximation is the GW approximation, so called from the mathematical form the self-energy takes as the product Σ = GW of the Greens function G and the dynamically screened interaction W. This approach is more pertinent when addressing the calculation of band plots (and also quantities beyond, such as the spectral function) and can also be formulated in a completely ab initio' way. The GW approximation seems to provide band gaps of insulators and semiconductors in agreement with experiment, and hence to correct the systematic DFT underestimation.