Summary form only given. Annular beam klystrons are well suited to high power operation. Significantly higher beam current can be transported than in conventional pencil beam klystrons (at the same beam voltage). Furthermore, we have designed cavity configurations that provide strong beam modulation and efficient energy extraction from the bunched electrons. Building on the pioneering work of Friedman and coworkers at the Naval Research Laboratory, we are developing high-power annular beam klystrons at L band (/spl sim/1.3 GHz) and X band (/spl sim/11.4 GHz). The goal of the L-band work is to generate a 1-sec-long microwave pulse at a power level of /spl sim/1 GW. The X-band source is designed to demonstrate that a triaxial geometry (grounded structures inside and outside the beam) allows efficient, high-power operation at significantly higher frequencies than an open beam tube geometry.