Drawing on the work Herman, Yoccoz, Khanin and others on circle diffeomorphisms I will try to motivate some questions about the generic behaviour of affine and generalised interval exchange transformations.

We will see that a baby case of this problem can be treated thanks to the use of moduli spaces of dilation surfaces. Dilation surfaces are essentially what one gets when gluing together parallel sides (of possibly different lengths) of Euclidean polygons.

Time permitting, we will try expose where the difficulty lies when one tries to generalise this approach to the general case.