  We give a survey of recent result regarding scaling limits of systems from
statistical mechanics, as well as the universality of the behaviour of such
systems in so-called cross-over regimes. It transpires that some of these
universal objects are described by singular stochastic PDEs. We then give a
survey of the recently developed theory of regularity structures which allows
to build these objects and to describe some of their properties. We place
particular emphasis on the renormalisation procedure required to give meaning
to these equations.
  These are expanded notes of the 20th Takagi lectures held at Tokyo University
on November 4, 2017.
