Abstract: I will describe some categories of "smooth spaces" which generalize the notion of manifold. The generalizations allow us to form smooth spaces consisting of subsets and quotients of manifolds, as well as loop spaces and other function spaces. In more technical language, these categories of smooth spaces are complete, cocomplete and cartesian closed. I will give examples, discuss possible applications and explain what we have learned about the homotopy theory of these categories. This is work in progress with Enxin Wu.