We'll give a survey on recent work with Jim Conant and Rob Schneiderman, formalizing higher order intersections of disks and spheres in 4-manifolds. In the easiest case where the manifold is a 4-ball, we developed an almost complete understanding of the relevant Whitney towers. It predicts new link invariants and gives new geometric interpretations of known link invariants, like Arf and Milnor invariants.