We consider the Kardar-Parisi-Zhang equation (KPZ) and the multiplicative Stochastic Heat Equation (SHE) in two space dimensions, driven by with space-time white noise. These singular PDEs are "critical" and lack a solution theory, so it is standard to consider regularized versions of these equations - e.g. convolving the noise with a smooth mollifier - and to investigate the behavior of the regularized solutions when the regularization is removed. Based on joint works with Rongfeng Sun and Nikos Zygouras, we show that these regularized solutions undergo a phase transition as the noise strength is varied on a logarithmic scale, with an explicit critical point. In the sub-critical regime, the regularized solutions of both KPZ and SHE exhibit so-called Edwards-Wilkinson fluctuations, i.e. they converge to the solution of the *additive* Stochastic Heat Equation (after centering and rescaling), with a non-trivial constant on the noise. We finally discuss the critical regime, where many questions are open.