Stochastic Hamilton-Jacobi equations appear naturally in a number of contexts, in particular in the level-set formulation of interface motions, when this motion is perturbed by random noise. The "irregular" time dependence of the right-hand side of these equations creates a number of mathematical difficulties. They are part of a class of equations ("fully nonlinear SPDE") which was introduced by Lions and Souganidis at the end of the 90s. Their work showed that one could extend the classical techniques of viscosity solutions to this context. In this talk, after presenting these equations, I will talk about various recent results concerning the long-time behavior of their solutions. Some of these results are only qualitative (convergence towards an equilibrium), but in some cases we can also prove quantitative results on the speed of convergence. We obtain in particular in the case of the movement of a graph by mean curvature that the stochastic term accelerates the convergence. Based on joint works with B. Gess, B. Seeger, P. Souganidis and P.L. Lions.