In heavy ion inertial fusion energy systems, intense beams of ions must be transported from the exit of the final focus magnet system through the target chamber to hit millimeter spot sizes on the target. In this paper, we examine three different modes of beam propagation: neutralized ballistic transport, assisted pinched transport, and self-pinched transport. The status of our understanding of these three modes is summarized, and the constraints imposed by beam propagation upon the chamber environment, as well as their compatibility with various chamber and target concepts, are considered. We conclude that, on the basis of our present understanding, there is a reasonable range of parameter space where beams can propagate in thick-liquid wall, wetted-wall, and dry-wall chambers.