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These lecture notes grew out of a series of lectures given by the secondnamed author in short courses in Toulouse, Matsumoto, and Darmstadt. The mainaim is to explain some aspects of the theory of "Regularity structures"developed recently by Hairer in arXiv:1303.5113 . This theory gives a way tostudy well-posedness for a class of stochastic PDEs that could not be treatedpreviously. Prominent examples include the KPZ equation as well as the dynamic$\Phi^4_3$ model. Such equations can be expanded into formal perturbativeexpansions. Roughly speaking the theory of regularity structures provides a wayto truncate this expansion after finitely many terms and to solve a fixed pointproblem for the "remainder". The key ingredient is a new notion of "regularity"which is based on the terms of this expansion.

Stochastic PDEs, Regularity Structures, and Interacting Particle Systems