Title: Whitney towers and Feynman diagrams in low-dimensional Topology. Abstract: The Whitney move plays a key role in many classification theorems for manifolds of dimension greater than 4, providing a link between algebraic data and geometric topology. Attempts to measure the failure of the Whitney move in dimension 4 lead to an obstruction theory defined in terms of towers of Whitney disks which are parametrized by untrivalent trees. I will describe applications of this approach to classical link theory and the study of immersed surfaces in 4-manifolds, as well as connections with the Kontsevich integral and finite-type invariants.