There are there great advances in the research on engineering strength theories in the latter half of the 20th Century. The first advance was the development of strength theory from the single -shear strength theory including the Tresca yield criterion and Mohr-Coulomb failure criterion to theoctahedral-shear strength theory; the second one was that from the octahedral -shear strength theory to the twin -shear strength theory; and the third was the theories from the single criteria to the unified strength theory. These three advances are summarized in this paper. It is interesting and useful for researchers to choose an appropriate failure criterion in studying the strength of materials and structures , for engineers to correctly use it and for students to understand strength theory. There are there great advances in the research on engineering strength theories in the latter half of the 20th Century. The first advance was the development of strength theory from the single-steal strength theory including the Tresca yield criterion and Mohr-Coulomb failure criterion to the octahedral -shear strength theory; the second one was that from the octahedral -shear strength theory to the twin -shear strength theory; and the third was the theories from the single criteria to the unified strength theory. These three advances are summarized in this paper. It is interesting and useful for researchers to choose an appropriate failure criterion in studying the strength of materials and structures, for engineers to correctly use it and for students to understand strength theory.