This course will focus on some singular stochastic PDEs (SPDEs) motivated by out-of-equilibrium statistical physics, in particular driven diffusive systems and stochastic interface growth. We will deal with equations that are "critical" or "super-critical", for which scaling or Renormalization group arguments suggest a Gaussian limit for large space-time scales (with logarithmic corrections to diffusivity in the critical dimension). In particular, this class includes the "Anisotropic KPZ equation" in dimension d=2 and the stochastic Burgers equation in dimension . We will explain how a careful analysis of the generator of the processes allows to prove Gaussian scaling limits in the super-critical case, and in the critical case in the weak-coupling regime.