In this talk we consider McKean-Vlasov Partial Differential Equations (PDEs), obtained as mean-field limit of interacting particle systems (in the limit when the number of particles grows). It is well-known that, even when the particle system has a unique invariant measure, the limit PDE may admit several equilibria, for example when the diffusion coefficient becomes too small in a non-convex landscape. The focus of the talk is to understand how the presence of common noise affects the stability of the limit equation. We will first present the known results for the case without common noise and then discuss how the introduction of common noise can enhance stability in certain respects.