In elastic-plastic structures subjected to dynamic external actions, if unbounded plastic deformations are developed, either local failure due to plastic fatigue (alternating plasticity) or gradual divergence of the deformed configuration (incremental collapse) will occur. Therefore, the boundedness in time of total plastic strains, and hence of total plastic work (usually referred to as adaptation or shakedown) is necessary for structural safety, in the sense that it rules out the occurrence of the above critical phenomena. Necessary and sufficient conditions for shakedown have been established by several authors. However, in many instances adaptation is not sufficient to ensure safety. In fact, even if plastic deformations can be proved to be finite, they can exceed some critical limit or exhaust the material ductility. In particular, for dynamic loading histories that cease after a certain time, a structure will certainly shakedown under any load amplitude, so that a safety criterion based on this event is clearly meaningless. Typical histories of this kind are earthquake or blast loadings. When the loading history is known, it is possible, in principle, to assess safety by following the actual plasto-dynamic evolution of the system, but this is often a laborious task and in several cases it provides far more information than is actually needed. On this remark rests the interest of methods capable to provide some essential information, such as upper bounds on maximum deflections or strains, through a moderate computational effort. In recent years several alternative techniques have been developed to bound from the above various quantities of interest. With the exception of very simple situations, the best bounds that can be obtained through any one of these methods involve the solution of constrained optimization problems. In this paper a study of several deformation bounding techniques is performed. The problem is formulated and the main previous results are outlined first with reference to general continua made of hardening materials. Then a class of discrete structural models (such as some finite element discretizations) is considered and, on this basis, two categories of deformation bounding techniques are described from the previous main results. All these techniques, some of which are new, permit the optimization of the upper bound by solving one or more mathematical programming problems of special forms. Some of the bounding procedures are shown to have merely theoretical interest, since they lead to cumbersome numerical procedures or to very coarse bounds. The formulations that appear to have practical application are compared from various standpoints (type of loading history, different hardening rules, influence of second order geometric effects, quantities to be bounded) and first assessment of their practical usefulness is attempted. Generalizations to second-order geometric and thermal effects and to situations in which the time history is not completely known are envisaged.