I will explain how three integration techniques for producing cohomology classes — Chen integrals for loop spaces, Bott-Taubes integrals for knots and links, and Kontsevich integrals for configuration spaces — come together in the computation of the cohomology of spaces of braids. The relationship between various integrals is encoded by certain graph complexes. I will also talk about the generalizations to other spaces of maps into configuration spaces (of which braids are an example). This will lead to connections to spaces of link maps and, from there, to other topics such as rope length, manifold calculus of functors, and a conjecture of Koschorke, all of which I will touch upon briefly. This is joint work with Rafal Komendarczyk and Robin Koytcheff.