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The phenomenon known as metastability has been widely studied for a large class of evolutive PDEs. From a general point of view, a metastable behavior appears when solutions to a PDE exhibit a first time scale in which they are close to some non-stationary state for an exponentially long time before converging to their asymptotic limit. In particular, through a transient process, a pattern of internal layers is formed from initial data over a short time interval; once this pattern is formed, the subsequent motion of the internal layers towards the steady state is exponentially slow. In this book we describe a general strategy to approach the problem of the slow motion of internal layers for a class of parabolic-hyperbolic systems, including viscous scalar conservation laws and the Jin-Xin system.