Monolithic and sandwich structures have been widely used as energy absorption structures in military and civil engineering. This article reviews theoretical analyses of monolithic beams and plates subjected to static loading, impulsive loading and impact by a mass systematically. Experimental data collected from the literatures are compared with these theoretical results. In addition, the critical impulses for the failure of the monolithic structures are also reviewed. Furthermore, sandwich structures under quasi-static, low-velocity impact, high-velocity impact and blast loading, as well as their failure modes, are also summarized. The research methodology involves experimental investigations, theoretical analyses and numerical simulations.