There is a very strong connection between the theory of flat surfaces and the theory of stability structures on categories. Namely, the moduli space of flat metrics on a surface is recovered from an algebraic gadget attached to the surface (its Fukaya category) by passing to the space of stability structures. There are many more spaces of stability structures which do not come from surfaces, but still (conjecturally) from geometry. I will discuss what to expect for them. No knowledge of the theory of stability structures will be assumed. References: arXiv:1409.8611, arXiv:1808.06364