Mars is covered with a layer of dust that has been homogenized by global dust storms. Dust, levitated by these storms as well as by the frequent dust devils, is the dominant weather phenomenon on Mars. NASA's Mars exploration rovers have shown that atmospheric dust falling on solar panels can decrease their efficiency to the point of rendering the rover unusable. Dust covering the surface of the moon is expected to be electrostatically charged due to the solar wind, cosmic rays, and the solar radiation itself through the photoelectric effect. Electrostatically charged dust has a large tendency to adhere to surfaces. The Apollo missions to the moon showed that lunar dust adhesion can hinder manned and unmanned exploration activities. In this paper, we report on our efforts to develop an electrodynamic dust shield to prevent the accumulation of dust on surfaces and to remove dust already adhering to those surfaces. The technology uses electrostatic and dielectrophoretic forces to carry dust particles off surfaces and to generate an electrodynamic shield that prevents further accumulation of dust. The concept of the electrodynamic dust shield was introduced by NASA in the late 1960s and later reduced to practice during the 1970s for terrestrial applications. In 2003, our laboratory, in collaboration with several universities, applied this technology to space applications, specifically to remove dust from solar panels on Mars. We show how, with an appropriate design, we can prevent the electrostatic breakdown at the low Martian atmospheric pressures. We are also able to show that uncharged dust can be lifted and removed from surfaces under simulated Martian environmental conditions. We have also been able to develop a version of the electrodynamic dust shield working under hard vacuum conditions that simulate the lunar environment. We have implemented the electrodynamic dust shield on solar arrays, optical systems, spectrometers, viewports, thermal radiators, batteries, and power systems, as well as on fabrics for spacesuits. We present data on the design and optimization of the electrodynamic dust shields and provide data on the performance of the different implementations of the technology for lunar and Martian exploration activities.