I will talk about spaces whose points parametrize certain kinds of orders in number fields, and ideal classes in those orders. The spaces give concrete classifications of these number theoretic objects. In particular, I will discuss the rings and ideals parametrized by binary forms, and concrete moduli spaces for the ideal classes of these rings. I will also discuss how we can understand these and other similar parametrizations geometrically, and when the integers are replaced by an arbitrary base scheme.