Several classes of mathematical theories of plasticity for work-hardening materials are surveyed and their advantages, disadvantages, and agreement with experiment discussed. Consideration is given to the proper correlation of tests on thin-walled tubes subjected to tension, torsion, and internal pressure in fixed but arbitrary ratio. The continuing debate between octahedral-shearing-stress and maximum-shearing-stress criteria of plastic deformation is re-examined and the more general alternatives are restated. Through an analysis of Osgood’s experimental results, it is made apparent that the more general point of view is required for the best correlation. A series of experiments are outlined which make the distinction between the various criteria of loading or deformation very large instead of just a few per cent as in previous work. In the evaluation of present mathematical theories it is shown that incremental-strain theories avoid obvious drawbacks of the so-called deformation type of theory. The concept of isotropic work hardening, assumed in practically all stress-strain relations, is explored and generalized. Strong limitations indicated by the Bauschinger effect, which cannot appear properly in such theories, are pointed out.