We consider a class of stochastic processes, which can be regarded as the perturbation of deterministic dynamics in a potential field. These processes exhibit a phenomenon known as metastable behavior if the potential field has several local minima. Metastable behavior is the phenomenon in which a process starting from one of local minima arrives at the neighborhood of the global minimum after a sufficiently long time scale. The precise asymptotic analysis of this transition time has been known only for the reversible dynamics, based on the potential theory of reversible Markov processes. In this presentation, we review this metastability theory for reversible dynamics, and introduce our recent generalization of this theory for non-reversible processes. (joint work with C. Landim)