  These notes have been prepared for a series of lectures given at the Sarajevo
Stochastic Analysis Winter School, from January 28 to February 1, 2019. There
already exist several excellent lecture notes and reviews on the subject, such
as (Hairer 2009) on (non-singular) stochastic PDEs, and (Hairer 2015, Chandra
and Weber 2017) on singular stochastic PDEs and regularity structures. The
present notes have two main specificities. The first one is that they focus on
a particular example, the Allen-Cahn equation, which allows to introduce
several of the difficulties of the theory in a gradual way, by increasing the
space dimension step by step. The hope is that while this limits the generality
of the theory presented, this limitation is more that made up by a gain in
clarity. The second specific aspect of these notes is that they go beyond
existence and uniqueness of solutions, by covering a few recent results on
convergence to equilibrium and metastability in these system.
