Stochastic electrodynamics is the Brownian motion of a charged particle in a random electromagnetic field with spectrum proportional to kω3 coth(kω/2kT). If the deterministic force field is simple harmonic, the properties of the system resemble closely those of the quantum-mechanical oscillator, but the extension to non-linear systems has proved to be a complex problem. The Kepler problem is an important test for stochastic electrodynamics, and there is some evidence that the same S0(4) symmetry which simplifies the treatments of both the classical and quantum-mechanical systems also reduces the complexity of the problem in stochastic electrodynamics.