Galaxy clusters are the most massive virialized structures in the local Universe. They often contain a prominent “brightest cluster galaxy” (BCG), surrounded by a plethora of — mostly red, early-type — satellite galaxies and a faint halo of intra-cluster light (ICL). It is generally thought that these satellites are strongly affected by their special environment: for example, their star formation activity is suppressed, and tidal forces strip or even completely disrupted them into the BCG or ICL. In my talk, I will present predictions from the Hydrangea cluster simulations on the nature of these processes. In the simulations, new cluster satellites are efficiently stripped of their gas, with both resemblances and tensions compared to recent observations. In contrast, the stellar mass of simulated satellites is largely unaffected by tidal stripping in clusters, so that almost all satellites accreted since redshift one still exist at the present day. I will discuss what this implies for the origin and buildup of stars in brightest cluster galaxies and the intra-cluster light, and the impact of the cluster environment on galaxy morphology.