This article is a study of the research development of electrodynamic phenomena occurring in ultra-fast electrodynamic drives. These types of linear drives are among the fastest, not only because of the huge accelerations achieved, but, above all, because of the extremely short reaction time. For this reason, electrodynamic drives are used in hybrid short-circuit breakers. The phenomena occurring in this type of drive are actually magneto-thermo-elastic in nature, but the coupling of these phenomena should be considered weak if the criteria for repeatable operation in a hybrid circuit breaker system are met. The authors have been researching this type of drive for many years through not only experimental studies, but also primarily simulation studies developing models of such drives. The authors present the history of the development starting from the first works of Thomson, and Kesserling and ending with the most current models, including mainly their own. This article presents mainly works studying electrodynamic phenomena. Thermal and mechanical phenomena were comprehensively presented by the authors in previous papers.