We proposed in Beltrán and Landim (J. Stat. Phys. 140:1065-1114, 2010) a new approach to prove the metastable behavior of reversible dynamics based on potential theory and local ergodicity. In this article we extend this theory to nonreversible dynamics based on the Dirichlet principle proved in Gaudillière and Landim (arXiv:1111.2445, 2011). We also include in this article the proof of the metastability of a class of birth and death chains. © 2012 Springer Science+Business Media New York.