  The goal of the present paper is to establish a framework which allows to
rigorously determine the large-scale Gaussian fluctuations for a class of
singular SPDEs at and above criticality, and therefore beyond the range of
applicability of pathwise techniques, such as the theory of Regularity
Structures. To this purpose, we focus on a $d$-dimensional generalization of
the Stochastic Burgers equation (SBE) introduced in [H. van Beijeren, R. Kutner
and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985]. In both
the critical $d=2$ and super-critical $d\geq 3$ cases, we show that the scaling
limit of (the regularised) SBE is given by a stochastic heat equation with
non-trivially renormalised coefficient, introducing a set of tools that we
expect to be applicable more widely. For $d\ge3$ the scaling adopted is the
classical diffusive one, while in $d=2$ it is the weak coupling scaling which
corresponds to tuning down the strength of the interaction in a scale-dependent
way.
