Abstract：

Many moduli spaces can be described as ball quotients. Examples include the Deligne-Mostow varieties, moduli of cubic surfaces and certain moduli spaces of lattice-polarized $K3$ surfaces. Here I will discuss the geometry of some of these examples, including their topology and different (partial) resolutions. I will also comment on the relationship with the Minimal Model Program. This talked is based on joint work with S. Casalaina-Martin, S. Grushewsky, R. Laza and Y. Maeda.