An outline of the continuous chain model for the description of the tensile and compressive deformation of polymer fibers below the glass transition temperature is presented. The basic mechanism is the contraction of the chain orientation distribution as a result of elastic, plastic, and viscoelastic shear deformation. The deformation of the fiber is calculated for finite strains and arbitrary values of the orientation parameter. The model explains the yield and the compressive strength of polymer fibers. The tensile curve, including yielding, is described in terms of the modulus for shear between the chains, the chain modulus, the chain orientation distribution, and a yield parameter. The response to complex time-dependent loading schemes can be calculated by introduction of the Eyring reduced time model. The elastic tensile deformation of carbon fibers is described in terms of the classical series aggregate model. It is shown that the modulus for shear between the graphitic planes and the orientation distribution of these planes govern the tensile and compressive properties of carbon fibers. The high values of the shear modulus are attributed to some covalent bonding between the graphitic planes. A survey of the various models for the strength of polymer fibers is presented and a new model is discussed, which explains the failure envelope of polymer fibers.