In contrast to the Kohn-Sham density-functional method, many-body perturbation theory allows an exact calculation of photoemission spectra and quasiparticle excitations from first principles. The key element in this approach is the electronic self-energy, which incorporates all exchange and correlation effects. Practical applications in solid-state physics typically employ the so-called GW approximation, which describes the coupling of electrons to density fluctuations (plasmons) and leads to accurate results for a wide range of materials. As the self-energy is nonlocal, frequency-dependent and complex, its numerical evaluation, even within the GW approximation, still constitutes a major computational challenge. In this talk I review the central numerical tasks and discuss recent algorithmic developments. In particular, the GW space-time method exploits the finite range of the nonlocal correlation functions in real space and imaginary time, leading to an improved scaling with the system size. Calculations for complex systems like defects on semiconductor surfaces have thus become possible.