Electronic band structure is one of the most important intrinsic properties of a material, and is in particular crucial in electronic, photo-electronic and photo- catalytic applications. Kohn-Sham Density-functional theory (KS-DFT) within currently available local or semi-local approximations to the exchange-correlation energy functional is problematic for the description of electronic band structure. Many-body perturbation theory based on Green’s function (GF) provides a rigorous framework to describe excited-state properties of materials. The central ingredient of the GF-based many-body perturbation theory is the exchange- correlation self-energy, which accounts for all non-classical electron-electron interaction effects beyond the Hartree theory, and formally can be obtained by solving a set of complicated integro-differential equations, named Hedin’s equations. The GW approximation, in which the self-energy is simply a product of Green’s function and the screened Coulomb interaction (W), is currently the most accurate first-principles approach to describe electronic band structure properties of extended systems. Compared to KS-DFT, the computational efforts required for GW calculations are much larger. Various numerical techniques or approximations have been developed to apply GW for realistic systems. In this paper, we give an overview of the theory of first-principles Green’s function approach in the GW approximation and review the state of the art for the implementation of GW in different representations and with different treatment of the frequency dependence. It is hoped that further methodological developments will be inspired by this work so that the approach can be applied to more complicated and scientifically more interesting systems.