This article treats some aspects of the propagation of a highly energetic heavy ion beam through a fusion target chamber. The stability of this beam to various perturbations is especially emphasized, first for vacuum propagation, and then for propagation through a background plasma. The results are illustrated by detailed calculations for a beam carrying 3 kA of current and consisting of 70GeV, singly ionized uranium ions. We find that such a beam will propagate stably through the target chamber. As a second example we have applied some of our results to the recently proposed I53+ concept in which up to a hundred beams of 40GeV iodine ions each carrying 4.2 kA of current are directed to the target. Our conclusions for this case are less optimistic, as such beams are likely to be subject to space-charge problems due to the increased value of v/γ and are prone to instabilities of the two-stream type.