Metastability is a phenomenon that occurs in the dynamics of a multi-stable non-linear system subject to noise. In models of statistical mechanics, it is related to the behavior of macroscopic quantities in the vicinity of a first order phase transition. A signature of metastable systems is the existence of multiple, well separated time scales. While at short time scales the system appears to be in a quasi-equilibrium (metastable state), at longer time scales rapid transitions between meta-stable states occur which are induced by random fluctuations. The understanding of the quantitative aspects of dynamical phase tran-sitions is one of the basic problem encountered in physics. In the first part of the book, we develop the potential theoretic approach to the analysis of metastability with a particular focus on systems in which the entropy is relevant. This approach allows to derive sharp estimates on metastable transitions times and its distribution. In the second part, we apply this method to the random-field Curie-Weiss-Potts model.