Abstract: Strength theory deals with the yield and failure of materials under complex stress state. Strength theory includes the yield criteria and failure criteria as well as the multi-axial fatigue criteria, multi-axial creep condition and material models in computational mechanics and computer codes, etc. It is an important foundation for researches on the strength of materials and structures. Strength theory is widely used in physics, mechanics, material science and engineering including civil, mechanical, hydraulic, aeronautic and astronautic, etc. It is of great significance in theoretical research and engineering application, and is also very important for the effective utilization of materials. Particularly for design purposes, it is important that a reliable strength prediction be available for various combinations of multi-axial stresses. It is an interdisciplinary field where physicists, mechanical and civil engineers interact in a closed loop. Strength theory is a very unusual and wonderful subject. The object is very simple, but the problem is very complex. It is one of the earliest objects, but it is still remaining open up to now. Great considerable efforts have been devoted to the formulation of strength theories and to their correlation with test data, but no single model or criterion has emerged which is fully adequate. Hundreds of models or criteria were proposed. It seems as if an old Chinese word said: “let a hundred flowers bloom and a hundred schools of thought contend”. About 85 equations of failure criteria are presented in this paper. This paper presents a summary of the advances in strength theory (including yield criteria, failure criterion, etc.) of materials under complex stress. About 80 kinds of failure criteria are reviewed and the relationship among them is discussed. The criteria are categorized and guidelines on selection of reasonable and failure criteria in researches and engineering applications are offered. The prospects for strength theories in the near future is also briefly discussed. It includes true triaxial test and confining tri-axial test (conventional triaxial test or false triaxial test), economic effect of strength theory, failure criterion for anisotropic materials and composite materials, failure criterion for porous materials and multi-phase materials, strength theories for other special materials, relations with shakedown, polyaxial fatigue, creep, damage and fracture, corner singularity and computational implementation of strength theories in computer codes, strength theories of materials under different environment, the esthetics of strength theories and researches in strength theory at different scales. The advances in strength theories are summarized under the framework of continuum and engineering application.