Recent years have seen the advent of prototypical quantum simulators: synthetic quantum systems which can be used to investigate quantum many-body physics in controlled conditions. Their tremendous potential is however constrained by our capability to accurately control them and to understand their intrinsically non-equilibrium behaviour, which poses formidable challenges. I will present ideas we are pursuing in this area. I will start by discussing an efficient approach to use state-of-the art simulators to realize interacting topological systems, whose excitations behave like exotic quantum particles (known as anyons). I will then discuss how low-energy states of such systems can be prepared and stabilized with the help of controlled dissipative effects. Finally, I will propose a novel non-equilibrium phenomenon, where temporal modulations of an anyon system lead to the emergence of more complex anyon classes, of interest for topological quantum computing.