  We prove universality of a macroscopic behavior of solutions of a large class
of semi-linear parabolic SPDEs on $\mathbb{R}_+\times\mathbb{T}$ with
fractional Laplacian $(-\Delta)^{\sigma/2}$, additive noise and polynomial
non-linearity, where $\mathbb{T}$ is the $d$-dimensional torus. We consider the
weakly non-linear regime and not necessarily Gaussian noises which are
stationary, centered, sufficiently regular and satisfy some integrability and
mixing conditions. We prove that the macroscopic scaling limit exists and has a
universal law characterized by parameters of the relevant perturbations of the
linear equation. We develop a new solution theory for singular SPDEs of the
above-mentioned form using the Wilsonian renormalization group theory and the
Polchinski flow equation. In particular, in the case of $d=4$ and the cubic
non-linearity our analysis covers the whole sub-critical regime $\sigma>2$. Our
technique avoids completely all the algebraic and combinatorial problems
arising in different approaches.
