Electroadhesion (EA) is an electrically controllable adhesion mechanism that has been studied and used in fields including active adhesion and attachment, robotic gripping, robotic crawling and climbing, and haptics, for over a century. This is because EA technologies, compared to other existing adhesion solutions, facilitate systems with enhanced adaptability (EA is effective on a wide of range of materials and surfaces), reduced system complexity (EA systems are both mechanically and electrically simpler), low energy consumption, and less-damaging to materials (EA, combined with soft materials, can be used to lift delicate objects). In this survey, we comprehensively detail the working principle, modeling, design, fabrication, characterization, and applications of EA technologies employed in robotics, aiming to provide guidance and offer potential insights for future EA researchers and applicants. Joint and collaborative efforts are still required to promote the in-depth understanding and mature employment of this promising adhesion and gripping technology in various robotic applications.