An analytical review of research since the 50s of the XXth century is presented, related to the development of stamping by explosion - a promising method of forming large hard-to-process parts, with the following results is established. The source of the force during explosion stamping is the high explosive charge. It is an important factor of criticalities during plastic deformation of a titanium alloy workpiece. There can be many reasons of their origin, and they are taken into account by researchers, but separately. At the same time, the process of force generation is quite complex, especially in the aquatic medium of the basin. However, the process of explosion stamping must be controlled under any conditions. Force delivery to the workpiece surface takes place during explosion stamping within water transfer medium in the basin. Compared with the air medium, the power of this effect increases significantly due to more complete use of the water energy capabilities. Most researchers agreed that the force effect, consisting of a shock wave, motion of a hydraulic flow and a gas bubble, result in complexity of mathematical modeling of the process of its displacement. Mechanical and geometric characteristics of the workpiece affect quality of the part made by explosion stamping. It is determined by the material ability to be plastically deformed and has the effect in strength, accuracy of geometric and angular dimensions, as well as surface condition of the part, the absence of defects, fractures and corrugations. At the same time, a unified quality assessment of the workpiece and part has not been established yet. The multiparametric process of plastic deformation during explosion stamping in water deforms not only of the workpiece, but also of the matrix, thus altering the part shape, walls thinning, the appearance of corrugations and flanges due to the complex stress state in various sections of the part. It is noted that the properties revealed during the explosion stamping, leading to change in corresponding parameters and characteristics, collectively determine the system to which systematic approach may be applied. A feature of explosion stamping study is the need for an experiment, the success of which is largely determined by the experience of the researcher. This stage of the explosion stamping evolutionary development is associated with the introduction of computer technologies into the practice of technological processes design. Research is mainly related to the use of object-oriented methods, the development and application of mathematical models that allow automating procedures for determining the parameters of the stamping process mode, followed by modeling individual actions during stamping by explosion. At the same time, it is necessary to evaluate the performance of stamping by explosion both as a whole and individual stages of its implementation.