In the introductory lecture, I will discuss a bit of statistical physics background/motivations about "driven diffusive systems", and explain how they can be mathematically modeled, either via discrete Markov chains (interacting particle systems, like the "Asymmetric Simple Exclusion Process") or via stochastic PDEs (like the Stochastic Burgers equation). I will also give a panorama of expected and proven results about "large-scale Gaussian fluctuations in dimension d\ge2". This part will include no proofs or technical details. In the more technical lecture, I will focus on the stochastic Burgers equation (and similar stochastic PDEs) in the critical dimension d=2. I will discuss what is the "weak coupling limit" and I will formulate a theorem (based on 2 joint recent works ( arXiv:2304.05730, arXiv:2108.09046) with G. Cannizzaro, D. Erhard, M. Gubinelli in different combinations) that says that these equations, in the weak coupling regime, have a Gaussian scaling limit at large scales. I plan to explain a bit the ideas behind the proof.