One of the most important tasks in the design process is to determine the material to be used. Therefore, it is important to know the stress limit state at which the material will fail and the nature of the failure. The material can be brittle, so that failure is defined by fracture, or ductile, so that failure is defined by yielding. The fracture criterion and the yield criterion are defined on the basis of experimental results for materials subjected to uniaxial stress. Since the determination of the critical values of the principal stresses in the multiaxial stress state at which the material enters a dangerous state is very complex based on experiments, there was a need for strength theories that predict the mechanical properties of the material in the multiaxial stress state based on the mechanical properties of the material in the uniaxial stress state. In this thesis, six theories of strength are explained: maximum normal stress theory, maximum normal strain theory, maximum shear stress theory, maximum total strain energy per unit volume theory, maximum shear strain energy per unit volume theory and Mohr's modified shear stress theory. The way a material breaks depends not only on the type but also on the temperature, the amount of stress, the chemical influences and the way the material was formed and manufactured, so that the same theory cannot always be applied to a particular material. In all theories, the normal and shear stresses at critical points are calculated first, followed by the calculation of the principal stresses. These stresses are then used to define the equivalent stresses that determine the strength conditions.