Structural elements, which deform inelastically, are often used in energy-absorbing devices due to their simple design and the high efficiency achieved by several buckling deformation mechanisms. The application of light ductile materials in transportation systems and increased loading intensity requires studies on the influence of the rate of loading and material characteristics on dynamic buckling behavior. The present review article is focused on summarizing the state of the art related to the inelastic dynamic stability and postbuckling behavior of various basic structural members. In particular, studies on the dynamic response of axially loaded idealized elastic-plastic models, rods, shells with circular and square cross sections, and long tubes are discussed with consideration given to the influence of the geometric and material characteristics as well as the loading conditions on the buckling phenomena observed in these structural elements. The findings from the theoretical and experimental investigations on the phenomenon of dynamic inelastic buckling reported in this review article are based on 118 references.