We consider a stochastic fourth-order degenerate-parabolic PDE describing the spreading of viscous thin films under the influence of thermal fluctuations. This equation has been suggested in the physical literature approximately 15 years ago independently by Davidovitch, Moro and Stone, and by Grün, Mecke and Rauscher. In this talk, we demonstrate how weak (martingale) solutions can be constructed in the case of linear and nonlinear gradient noise. Key ingredients are (combined) energy and entropy estimates in conjunction with a tailor-made approximation procedure in order to control the formation of shocks. The talk is based on joint works with Konstantinos Dareiotis (University of Leeds), Benjamin Gess (Bielefeld University and Max Planck Institute, Leipzig) and Günther Grün (University of Erlangen-Nuremberg).