Physical systems with unexpected, or `hidden,’ symmetries have often played an important role in physics, beginning with the classical Kepler problem whose Laplace-Runge-Lenz vector ensures the closure of planetary orbits, and degeneracies of the Hydrogen spectrum. I will describe how precisely the same symmetry governs a unique four-dimensional quantum field theory, a maximally supersymmetric (`N=4') cousin of the strong-interaction Yang-Mills theory. After reviewing progress in recent years in using these symmetries to solve this model, I will describe novel applications involving massive particles. Combining the Laplace-Runge-Lenz vector with relativity then yields a novel way to calculate the spectrum of its Hydrogen-like bound states, including relativistic corrections. Based on 1408.0296.