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 also present a method to provide sharp characterizations of the spectrum for metastable systems on locally finite graphs We also briefly discuss the very related phenomena of phase-separation for the so-called exchange-driven growth model, which is the mean-field of the cluster dynamic in zero-range and related interacting particle systems. For this model we also prove for a suitbale choice of interaction kernels dynamic self-similar behavior which prove is based on a connection to a discrete Bessel-type process.