Much of my work is concerned with the regularity of harmonic maps into certain classes of singular spaces. In my first talk, I will discuss the history of (and motivation for) studying harmonic maps into singular spaces of nonpositive curvature. In particular, we will define the singular sets of some of these maps, and sketch a proof that the singular set is "small." The second talk will discuss a method for showing "stratification" results for these singular sets, giving a more precise analysis of the "smallness" of the set. This second talk is based on joint work with Christine Breiner.