We consider the dynamics of several vortex rings in an ideal incompressible fluid governed by the 3D Euler equations. While the self-induced motion of a single vortex ring amounts to a translation along its symmetry axis, Helmholtz conjectured in 1858 that coaxial vortex rings may exhibit the so-called leapfrogging dynamics. Specifically, the vortex rings display a periodic motion in which they repeatedly pass through each other by shrinking and widening respectively due to mutual interactions. The phenomenon is well observed numerically and experimentally. We rigorously justify these dynamics on a logarithmic time scale for a general class of initial data corresponding to N sharply concentrated coaxial vortex rings. In order for the leapfrogging to occur, the vortex rings need to be placed at a very small mutual distance leading to singular interactions. The key mathematical difficulty then consists in proving that the sharp concentration of vorticity persists for positive times. We provide new and improved estimates on the localization of the vorticity in strong and weak sense. In particular, this allows us to derive the respective asymptotic motion law. Joint work with M. Donati (Grenoble), C. Lacave (Chambery) and E. Miot (Grenoble).