One of the distinctive properties of composite materials is their anisotropy, which allows, in particular, to use of their strength properties maximum. Thanks to new technologies, in particular 3-D printing, there are new opportunities for purposeful formation of the structure of composite materials and their properties. An important issue here is ensuring strength, which is determined depending on the structure of the composite as a whole and the mechanical characteristics of its individual components. The article reviews well-known strength criteria (Goldenblatt-Kopnov, Mises-Hill, Fisher, Hoffman, Tsai-Wu, and others), the relationship between them, and data on the reliability of their application under certain load conditions. Some of them assume the same properties of materials under conditions of compression and tension and are obtained by generalizing the classical theories of strength and plasticity of isotropic bodies. The criteria that take into account the different resistance of the composite material to tension and compression, which is inherent in many composite materials, have gained greater practical application. When choosing a strength criterion, from the point of view of structure, it is worth considering separately layered and reinforced composite materials, and from the point of view of load - the static or cyclic nature of its application. The criteria listed above do not take into account the damage accumulation in the material that occurs under the influence of external loads. The number of such known criteria is limited, because today there is no method for predicting the stage of scattered fracture of composite materials taking into account the anisotropy of mechanical properties, and the choice of the damage parameter is not justified. The perspective of developing such criteria for clarifying the received conclusions regarding the load-bearing capacity of products made of composite materials under different types of application of external loads is noted.