The primary limit to the transverse densities achievable in a low energy hadron accelerator is due to space-charge forces at injection. Values of the Laslett tune shift parameter achieved in existing accelerators range from 0.4 to 0.9. Our understanding of the phenomona is still rudimentary but developing rapidly with the help of simulations. This limit affects essentially all hadron accelerator complexes. The cures include raising injection energies, increasing the degree of cascading in the complex, and lengthening the bunches.

Transition crossing represents an important limitation in the achievement of high longitudinal phase-space densities in circular accelerators. A variety of effects need to be contended with while passing through transition, including the microwave instability, space-charge, and the Johnsen effects. These are exacerbated by the tendency of the bunch length to approach zero, accompanied by a loss of Landau damping as transition is approached. Transition crossing impacts essentially all proton accelerator complexes, and even the high energy storage ring RHIC. Potential cures include avoidance, yt jumps, al control, and higher harmonic cavities.