First, we give a brief introduction to critical singular SPDEs. As a particular example, we review the 2D (linear) multiplicative stochastic heat equation, which is closely related to the study of the directed random polymer model. The classical approach of performing a Picard iteration of the solution, yields an infinite series of stochastic iterated integrals. In contrast to considering subcritical SPDEs, each term in the infinite expansion/series has a positive contribution to the solution. In the second part of the talk, we treat the (critical) 2D Allen-Cahn equation with white noise initial datum in a weak coupling regime. We will present an approach that keeps track of each occuring term in the Picard iteration, using the notion of rooted trees and determine their non-trivial Gaussian fluctuations exactly. Furthermore, by exploiting the structure of the equation, we can approximate the infinite series while controlling the imposed error, and determine its limiting law. This is based on joint work with Tommaso Rosati and Nikos Zygouras.