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, three different modes of beam propagation are examined: neutralized ballistic transport, assisted pinched transport, and self-pinched transport. The status of the authors' 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. It is concluded that on the basis of the present understanding, there is a reasonable range of parameter space where beams can propagate in thick-liquid-wall, wetted-wall, and dry-wall chambers.