Abstract: The study of advanced methods of acceleration has been largely concentrated on two quite different areas of applicability — first, the acceleration of electrons and positrons to ultra-relativistic energies (> 1 TeV) and second, the acceleration of protons and ions from rest or very low energies to a speed that is a significant fraction of the speed of light. Both applications share the common goal of seeking to achieve accelerating gradients far in excess of those obtainable with conventional accelerating structures, e.g., E ≫ 10 MV/m.