Some real and unreal enumerative geometry Frank Sottile The geometric counterpart of looking for real solutions to a system of polynomial equations is real enumerative geometry, which addresses the following question of Fulton: "How many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures". Quite surprisingly, in every known case it is possible to have all solutions be real. Such an enumerative problem is called fully real. Recent progress in this area has been stimulated by a remarkable conjecture of Shapiro and Shapiro, which is false in its original generality. This talk will discuss some totally real enumerative problems, the conjecture of Shapiro and Shapiro, and recent progress in this area. The main goal will be to describe refinements and extensions of the conjecture of Shapiro and Shapiro, which are supported both by extensive experimental evidence, and some theorems. In particular, we will describe a family of `fully unreal' enumerative problems for which it is possible to have *no* real solutions.