Abstract: 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 the various integrals is encoded by graph complexes. This will lead to various connections to the topology of spaces of link maps, manifold calculus, and rope length of knots. This is joint work with Rafal Komendarczyk and Robin Koytcheff.