Trying to give a proper definition of the KPZ (Kardar-Parisi-Zhang) equation in dimension \(d \ge 3\) is a challenging question. A plan to do so, is to first consider the well-defined KPZ equation when the white noise is smoothed in space. For \(d \ge 3\) and small noise intensity, the solution is known to converge to some random variable as the smoothing is removed. It is also known that the limiting random variable can be related to the free energy of a Brownian polymer, in a smoothed white noise environment. In this talk, we will state some recent results about the fluctuations of the convergence of the solution. In particular, we will show that the fluctuation of the solution, around the rescaled free energy of the polymer, converges pointwise towards a Gaussian random variable (joint work with Francis Comets and Chiranjib Mukherjee).