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A fundamental notion in statistical mechanics is phase transition: a microscopic system composed of a huge number of random particles depends on a thermodynamic parameter, and the system undergoes sudden changes in its large-scale structure as this parameter varies across a critical point. Self-avoiding walk was introduced in the 1940s as a model in chemistry of a long chain of molecules, and is now viewed as a fundamental model in the rigorous theory of statistical mechanics. By introducing a positive parameter which provides a penalty to self-avoiding walk which is exponential in the walk's length, we obtain an example of phase transition. Recent work with Hugo Duminil-Copin shows that uniformly chosen self-avoiding walks of given high length move sub-ballistically, and this is related to the nature of this phasetransition at the critical point. I will give an overview of the main elements of the proof. Considering instead subcritical self-avoiding walk, and focussing on the planar case, we obtain a natural model for the problem of phase separation: when one substance is suspended in another, such as oil in water, a droplet forms, whose boundary approximates a smooth profile predicted by Wulff. Modelling the problem using a planar model such as subcritical self-avoiding walk, the fluctuation of the droplet boundary from its typical profile exhibits characteristic scaling exponents - 2/3 longitudinally and 1/3 latitudinally - which I derived a couple of years ago. The behaviour arises from a competition of local Gaussian randomness and global curvature constraints. Phase separation in this guise is a static model. However, the Gaussian competition with curvature, and the two exponents, are shared by many dynamic models, of interfaces growing at random and subject to forces of surface tension. These models form the Kardar-Parisi-Zhang universality class. Resampling techniques from the phase separationpapers find counterparts in more recent work, joint with Ivan Corwin, in which a natural Gibbs property is proved for the multi-line Airy process, which is a fundamental scaling limit encountered in KPZ universality. This Brownian-Gibbs property is valuable in, for example, improving regularity assertions about the Airy process. These ideas form the subject of the final part of the talk.