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These notes have been prepared for a series of lectures given at the SarajevoStochastic Analysis Winter School, from January 28 to February 1, 2019. Therealready exist several excellent lecture notes and reviews on the subject, suchas (Hairer 2009) on (non-singular) stochastic PDEs, and (Hairer 2015, Chandraand Weber 2017) on singular stochastic PDEs and regularity structures. Thepresent notes have two main specificities. The first one is that they focus ona particular example, the Allen-Cahn equation, which allows to introduceseveral of the difficulties of the theory in a gradual way, by increasing thespace dimension step by step. The hope is that while this limits the generalityof the theory presented, this limitation is more that made up by a gain inclarity. The second specific aspect of these notes is that they go beyondexistence and uniqueness of solutions, by covering a few recent results onconvergence to equilibrium and metastability in these system.

An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures