An overview is provided for the novel relativistic klystron amplifiers which are under active study at the Naval Research Laboratory. These amplifiers are driven by an annular intense relativistic electron beam (500-kV, 10-kA range), which is modulated by an external RF source (1.3-GHz, 100-kW range). Experiments, theory, simulation, and simple models are presented to illustrate the unusual properties of such devices which result from the intense space charge of the beam. Chief among them are electrostatic insulation against vacuum breakdown at high power levels, efficient current modulation, short bunching length, and amplitude and phase stability of the output signal. Many of these unexpected features were revealed in two separate experiments: one with a lower current beam (5 kA, 2-cm beam radius), and the other one with a higher current beam (16 kA, 6.6-cm beam radius). Three gigawatts of RF power at 1.3 GHz were generated with the large diameter beam at an efficiency of 35% with 37-dB gain. These experiments are reviewed, along with a combination of particle simulation results and analytic models which facilitate the interpretation. Special attention is paid to the unfamiliar features of these amplifiers, and the critical problems which must be solved before such amplifiers can fulfil their potential in a wide range of applications are addressed. >