We present results concerning the qualitative and quantitative description of interacting systems with particular emphasis on those possessing a phase transition under the change of relevant system parameters. For this, we first discuss and identify different kinds of phase transitions (continuous and discontinuous) for mean-field limits of interacting particle systems on continuous and discrete state spaces. Since phase transitions are intimately related to the metastability of the stochastic particle system, we show how a suitable mountain pass theorem in the space of probability measures can describe the metastable behaviour of the underlying finite particle system. We also show that the particle system close to a discontinuous phase transition shows coarsening where smaller clusters, which grow through coagulation events. We provide numerical experiments that those phenomena can be also observed by a SPDE of Dean-Kawasaki type consisting of the McKean-Vlasov equation with some suitable conservative noise. Furthermore, we propose a simplified description of the growth process, which might give insights into the relevant time-scales. joint works with José Carrillo, Rishabh Gvalani and Greg Pavliotis.