Betelgeuse, the bright red supergiant (RSG) in Orion, is a runaway star. Its supersonic motion through the interstellar medium has resulted in the formation of a bow shock, a cometary structure pointing in the direction of motion. We present the first 3D hydrodynamic simulations of the formation and evolution of Betelgeuse’s bow shock. We show that the bow shock morphology depends substantially on the growth timescale for Rayleigh-Taylor versus Kelvin-Helmholtz instabilities. We discuss our models in light of the recent Herschel, GALEX and VLA observations. If the mass in the bow shock shell is low (~few  × 10-3M⊙), as seems to be implied by the AKARI and Herschel observations, then Betelgeuse’s bow shock is very young and is unlikely to have reached a steady state. The circular, smooth bow shock shell is consistent with this conclusion. We further discuss the implications of our results, in particular, the possibility that Betelgeuse may have only recently entered the RSG phase.