Abstract <p>Currently, there are two theories of metal plasticity in solid mechanics: the classical mathematical theory of plasticity [1, 2] and the physical and mathematical theory of plasticity [3]. Both theories are used to solve applied problems of plasticity, in particular, in the development and improvement of technological processes of metal pressure treatment [5–7]. This paper examines the relationship between the above-mentioned theories. Based on the analysis of the initial postulates and principles, the disadvantages and advantages of the theories for solving problems of theoretical design of technological processes of metal pressure treatment are identified. The analysis allows us to assert that the physical and mathematical theory of plasticity is a more general theory, which includes the mathematical theory of plasticity as a special case and which does not have the disadvantages of the latter. It follows from this that the physical and mathematical theory of metal plasticity is an important scientific achievement in solid mechanics.</p>