The KPZ equation is a popular model of one-dimensional interface
propagation. From heuristic consideration, it is expected to be
"universal" in the sense that any "weakly asymmetric" or "weakly noisy"
microscopic model of interface propagation should converge to it if one
sends the asymmetry (resp. noise) to zero and simultaneously looks at
the interface at a suitable large scale. The only microscopic models for
which this has been proven so far all exhibit very particular that
allow to perform a microscopic equivalent to the Cole-Hopf transform.
The main bottleneck for generalisations to larger classes of models was
that until recently it was not even clear what it actually means to
solve the equation, other than via the Cole-Hopf transform. In this
talk, we will see that there exists a rather large class of continuous
models of interface propagation for which convergence to KPZ can be
proven rigorously. The main tool for both the proof of convergence and
the identification of the limit is the recently developed theory of
regularity structures, but with an interesting twist.