Observational and theoretical progress has established that active galactic nuclei (AGN) play a decisive role in the formation and evolution of individual galaxies as well as galaxy groups and clusters. In particular, there is compelling evidence that AGN vigorously interact with their surroundings in the central regions of clusters, indicating that any realistic model of cluster formation needs to account for these processes. This is also suggested by the failure of previous generations of hydrodynamical simulations without AGN feedback to simultaneously account for the paucity of strong cooling flows in clusters, the slope and amplitude of the observed cluster scaling relations, and the high-luminosity cut-off of central cluster galaxies. Here we use high-resolution cosmological simulations of a large cluster and group sample to study how AGN affect their host systems. We find that AGN feedback brings the halo gas fraction and the X-ray luminosity-temperature scaling relation, both of which are notoriously difficult to reproduce in self-consistent hydrodynamical simulations, into excellent agreement with observations. At the same time, the luminosities of central cluster galaxies and the ages of their stellar populations become much more realistic. Furthermore, our sample of clusters and groups is of high enough resolution to accurately resolve galaxy populations down to the smallest galaxies that significantly contribute to the stellar mass budget. It is therefore also perfectly suited to study the stripping of stars form cluster galaxies and the build-up of the intracluster light (ICL). We describe and test four different methods to identify the ICL in cluster simulations, thereby allowing us to assess the reliability of the measurements. For all of the methods, we consistently find a very significant ICL stellar fraction which exceeds the values typically inferred from observations. However, we show that this result is robust with respect to numerical resolution and integration accuracy, remarkably insensitive to changes in the star formation model, and almost independent of halo mass. We find that intracluster stars are preferentially stripped in a cluster’s densest region from massive galaxies that fall into the forming cluster at z > 1. Surprisingly, some of the intracluster stars also form in the intracluster medium inside cold gas clouds that are stripped out of infalling galaxies.