Failure criteria are of great importance in progressive damage simulation and ultimate strength prediction of composite materials. Towards the damage-failure problem of anisotropic composites, the existing failure criteria are comprehensively classified and summarized according to model architecture and development venation, with focus on the failure-mode-independent and failure-mode-dependent failure criteria. Meanwhile, the theoretical models based on the action-plane assumption were analyzed. Moreover, the basic theories and methods for constructing stress-invariant-based criteria were described, and the resultant models were presented. Finally, the other two types of failure criteria, i.e., strain energy criteria and damage-based criteria were discussed. The investigation shows that the rationality and applicability of the higher-order phenomenological failure criteria still need to be verified, and the foundation and main direction of the strength theory development for anisotropic composites is still the quadratic stress criterion; the energy criterion and the damage criterion show rationality and applicability for nonlinear and quasi-brittle composites, but their expressions are more complicated. On the basis of in-depth analysis and research, the development trend of strength theory of composites is prospected, and an important and difficult direction of "damage-fracture synergistic theory" is put forward, with a view to providing more valuable references for the design, evaluation and application of composites and their structures.