A theory of the tensile strength of oriented polymer fibres is presented. From an analysis of the observed fracture envelope it is shown that the criterion for fracture of the fibre is either a critical shear stress or a critical shear strain.Owing to the chain orientation distribution in the fibre, the initiation of fracture is likely to occur in domains whose symmetry axes have orientation angles in the tail of this distribution. By considering the fibre as a molecular composite, the tensile strength is calculated as a function of the modulus. The results are compared to the observed values of PET,POK,cellulose II,PpPTA,PBO and PIPD fibres. In addition, the relation between the ultimate strength and the chain length distribution is investigated. By using the critical shear strain as a fracture criterion in the Eyring reduced time model, relations are derived for the fibre strength as a function of the load rate, as well as for the lifetime under constant load.Moreover, this model predicts the dependence of the strength on the temperature. The theoretical relations are compared to the experimental results on PpPTA fibres.