We explain how the Edge-reinforced random walk, introduced by Coppersmith and Diaconis in 1986, is related to several models in statistical physics, namely the supersymmetric hyperbolic sigma model studied by Disertori, Spencer and Zirnbauer (2010), the random Schrödinger operator and Dynkin's isomorphism.

We also discuss recent non-reversible generalizations of the ERRW and the VRJP. Based on joint works (or work in progress) with C. Sabot, X. Zeng, T. Lupu, M. Disertori and S. Baccalado.