The CAT(0) inequality says that in non-positively curved spaces triangles are no fatter than the Euclidean ones. The richest examples are symmetric spaces of non-compact type, which carry Riemannian, CAT(0), and Lie group structures at once. Using SL(n,\R)/SO(n) as a running example, I will explain why flats are important, and how to describe points at infinity in those spaces. Then I'll say a bit about some work on boundaries of horospheres, and end with something hopefully interesting: dynamics on those spaces and how we use them to get counting results.