This paper investigates the onset of localized necking in thin sheets under biaxial stretching using a simplified constitutive model of a pointed vertex on the yield locus. The model predicts that localized necking occurs when the equations of deformation theory of rigid-plastic solids apply for fully-active stress increments. The predictions agree reasonably well with experimental observations. The results support the hypothesis that vertex formation on the yield locus under continued plastic flow is responsible for the onset of localized necking. The implications of this conclusion for the study of ductile fracture in solids as a material instability are far-reaching. However, explanations based on a smooth yield-locus but small initial inhomogeneities cannot be ruled out, and both initial imperfections and yield-vertex effects may contribute to localization instabilities. The paper discusses the conditions for the onset of localized necking in plane stress, the equations of deformation theory of plasticity as a model for flow-theory behavior under fully-active stress-increments at a yield vertex, and the analysis and numerical results for localized necking in thin sheets under biaxial stretching. The results show that localized necking occurs when the material is subjected to biaxial stretching, and the direction of the neck is determined by the stress state and the material properties. The paper also compares the predicted forming limit curves with experimental observations and shows that the results agree well with the experimental data. The paper concludes that the onset of localized necking in thin sheets under biaxial stretching can be explained as a bifurcation from a state of uniform deformation, and that the destabilizing effect of a pointed vertex on the yield locus is a key factor in this process. The results also show that the predicted forming limit curves agree well with experimental observations, and that the results are sensitive to the details of the stress-strain relations of the material.This paper investigates the onset of localized necking in thin sheets under biaxial stretching using a simplified constitutive model of a pointed vertex on the yield locus. The model predicts that localized necking occurs when the equations of deformation theory of rigid-plastic solids apply for fully-active stress increments. The predictions agree reasonably well with experimental observations. The results support the hypothesis that vertex formation on the yield locus under continued plastic flow is responsible for the onset of localized necking. The implications of this conclusion for the study of ductile fracture in solids as a material instability are far-reaching. However, explanations based on a smooth yield-locus but small initial inhomogeneities cannot be ruled out, and both initial imperfections and yield-vertex effects may contribute to localization instabilities. The paper discusses the conditions for the onset of localized necking in plane stress, the equations of deformation theory of plasticity as a model for flow-theory behavior under fully-active stress-increments at a yield vertex, and the analysis and numerical results for localized necking in thin sheets under biaxial stretching. The results show that localized necking occurs when the material is subjected to biaxial stretching, and the direction of the neck is determined by the stress state and the material properties. The paper also compares the predicted forming limit curves with experimental observations and shows that the results agree well with the experimental data. The paper concludes that the onset of localized necking in thin sheets under biaxial stretching can be explained as a bifurcation from a state of uniform deformation, and that the destabilizing effect of a pointed vertex on the yield locus is a key factor in this process. The results also show that the predicted forming limit curves agree well with experimental observations, and that the results are sensitive to the details of the stress-strain relations of the material.