The problem of many interacting electrons is hard to solve. Many-body perturbation theory is a set of techniques to reduce this problem to one of a single electron moving in an effective, energy-dependent potential. This potential is also called self-energy and can be represented as an infinite sum of terms describing the interaction of the single electron with the other electrons. In practice, one calculates only a few of these terms which are believed to be dominant for certain interaction patterns. The self-energy can be used to calculate the single-particle Green’s function whose poles can be identified with electron addition and removal energies. These can be probed in direct and inverse photo-emission spectroscopy. It also gives access to the 4-point vertex function which in turn can be used to calculate the interacting density-density response function whose poles describes the energies of electron-hole pairs. These can be probed in photo-absorption spectroscopy. Finally, electron-electron interaction energies can be obtained from the self-energy as well. Comparing the electron-electron interaction energies of different isomers is for instance useful to determine which isomer is most stable. The most important terms in the self-energy are the Hartree contribution, describing the interaction of the electron with the average charge density produced by the other electrons, as well as the exchange term, accounting for the Fermionic structure of the wave functions. These terms already account for the majority of electron-electron interactions. However, for a realistic description of many-electron systems, additional terms need to be considered. The combined effect of these terms is called correlation. In a canonical method due to Hedin, the correlation part of the self-energy is expanded in powers of the screened electron-electron interaction. The first-order term in this expansion is called GW term and truncating the self-energy after this term is called GW approximation. The GW approximation is a standard technique which is computationally feasible for systems with many hundreds of electrons. It rests on the assumption that the dominant source of electron-electron correlation is the screening of the electron-electron interaction by the presence of other electrons. This is a good approximation for processes which are dominated by long-ranged interactions, for instance when an electron is removed from a finite system. In case short-ranged interactions become important, additional terms, so-called vertex corrections, need to be considered. The G3W2 term is the next-to-leading order term in the expansion of the self-energy and it massively improves the description of short-range correlation. In this thesis, we have implemented these techniques for finite systems and assessed their precision and accuracy. Our implementation equips researchers with tools to describe spectroscopic properties of relatively large systems with good accuracy. We have demonstrated this via applications to the ionization of DNA oligomers as well as the low-lying excited states of the reaction center of photosystem II. A more detailed summary of the research we have performed including an outlook on future work can be found in the conclusions of this thesis.