Hitchin's integrable system is one of the fundamental examples of mirror symmetry in the sense of Strominger-Yau-Zaslow.

Many of the structures described in this workshop can be seen very concretely in this example. This example has moreover some extra structure, coming from the fact that it is actually a hyperkahler space.

I will review an approach to these moduli spaces which arose in my joint work with Davide Gaiotto and Greg Moore. The key player in the story is a set of cluster-type coordinate systems, very closely related to those appearing in the work of Fock-Goncharov. We will also see scattering diagrams very similar to those in the work of Gross-Hacking-Keel.

In the end I will describe a new point: when one tries to extend the cluster description over certain singular loci of the moduli space, the most natural description seems to involve not an ordinary cluster algebra, but rather a noncommutative version thereof. One special case of this story appears closely related to the "noncommutative marked surfaces" recently introduced by Berenstein-Retakh; there is also related work of Goncharov-Kontsevich.