AbstractThe charged-particle beams to be used for driving the targets in future heavy-ion inertial fusion reactors are strongly affected by space-charge forces. This is true for both the r.f. linear accelerator compressor ring scenario as well as for the induction accelerator system. The low current ( I≳ 1 A), long-pulse (about a millisecond) beams from several ion sources must be accelerated, merged and compressed longitudinally to achieve the peak power (about 20 kA or less; about 10 GeV or less; about 10 ns or less) required at the target. To achieve fuel ignition, the final beams must be focused to a small spot of only a few millimeters. This in turn requires careful control of the six-dimensional phase-space volume of the beams during the entire acceleration and final transport process. The major effects that can lead to increases in the transverse and longitudinal emittances will be reviewed, with particular emphasis on thermodynamic consideration. There has been significant progress in recent years on understanding the underlying beam physics. Some highlights of ongoing experimental and theoretical research at various laboratories will be presented.