In electrodynamics, the levitation ensues when there is a relative motion between a stator and a conductor owing to which high eddy currents are induced. The repulsive forces generated by eddy currents are large enough to carry the moving body passively without violating Earnshaw's criterion. Electrodynamic bearings (EDB) use this technique to levitate the rotor in axial or radial degrees of freedom. The present article reviews the literature on electrodynamic passive magnetic bearings to provide an in-depth examination of EDB modeling, analysis, and development. Different types of EDB configurations are highlighted, along with improvements in their technology. In addition, essential advancements in mathematical modeling, finite element techniques, and experiments used to determine the bearing characteristics in terms of accuracy are highlighted. Furthermore, critical developments in rotordynamic analysis in terms of high-speed application have been reviewed. Finally, advances in the application of EDB are discussed.