There is currently a controversy as to whether Voyager 1 has already crossed the termination Shock, the first boundary of the heliosphere. The region between the termination shock and the heliopause, the heliosheath, is one of the most unknown regions theoretically. In the heliosheath magnetic effects are crucial, as the solar magnetic field is compressed at the termination shock by the slowing flow. Recently, our simulations showed that the heliosheath presents remarkable dynamics, with turbulent flows and the presence of a jet flow at the current sheet that is unstable due to magnetohydrodynamic instabilities. In this paper we review these recent results, and present an additional simulation with constant neutral atom background. In this case the jet is still present but with reduced intensity. Further study, e.g., including neutrals and the tilt of the solar rotation from the magnetic axis, is required before we can definitively address how the heliosheath behaves. Already we can say that this region presents remarkable dynamics, with turbulent flows, indicating that the heliosheath might be very different from what we previously thought. © 2004 American Institute of Physics