This talk focuses on the height function of the six-vertex model that exhibits a roughening transition: depending
on the parameters, it is either localised or is expected (and sometimes proven) to converge to the Gaussian Free Field. We discuss both of these regimes:
- joint works with Moritz Dober and Sébastien Ott use the (localised) height function to show the invariance principle and wetting in the Potts model in 2D at Tc for all q > 4;
- a joint work with Lammers gives an elementary proof of delocalisation and deduces continuity of the phase transition in the Potts model when 1 \leq q \leq 4.

The key tools used in the proofs are positive correlation inequalities that allow to decouple different parts of the height function. There are also connections the Ashkin-Teller and its graphical representation.