The calculation of the ground-state and excited-state properties of materials is one of the main goals of condensed matter physics. While the most successful first-principles method, the density-functional theory (DFT), provides, in principle, the exact ground properties, the many-body method is the most suitable approach for studying excited-state properties of extended systems. Here we discuss general aspects of the Green\'s function and different approximations for the self-energy to solve the Dyson equation. Further we present some tools for solving the Dyson equation with several approximations for the self-energy: a highly precise combined basis method providing the band structure in the Kohn-Sham approximation, and some implementations for the random-phase approximation.