Homogenization of Hamilton-Jacobi equations with non-convex Hamiltonians in stationary ergodic random media is a largely open problem. In the last 5 years several classes of examples and counter-examples appeared in the literature. The majority of known examples concern inviscid Hamilton-Jacobi equations. We shall discuss two classes of viscous Hamilton-Jacobi equations with non-convex Hamiltonians in one space dimension for which homogenization holds and pose several open questions. The talk is based on joint works with Andrea Davini (Sapienza - Universita di Roma) and with Atilla Yilmaz (Temple University) and Ofer Zeitouni (Weizmann Institute and NYU).