Day #1: The lecture is loosely based on the book [F-Hairer, Course on Rough Paths, with an Introduction to Regularity Structures, Springer 2014] but I will aim for a balance between introductory/standard material and advanced topics taken from the 2nd edition (in preparation), hopefully making the talk accessible to a diverse audience. Among the new material I will introduce the rough volatility model from quantitative finance as simple (but not too simple) example of a regularity structures that exhibits many features also seen in the analysis of singular stochastic partial differential equations like KPZ.