A theoretical study of the stability of extending liquid filaments has been carried out. The interaction of surface tension and different fluid rheological properties is investigated. It is also hypothesized that cohesive failure or fracture will occur if a critical stress level is reached. It is predicted that viscosity and viscoelasticity tend to stabilize the filaments. However, even extremely high viscosity filaments will neck and exhibit ductile failure. In highly viscoelastic fluids, defects tend to heal during stretch. Highly viscoelastic fluid filaments fail by fracture. The theory is used to predict the failure of molten polymer filaments as a function of molecular weight. The extensibility or spinnability of filaments is predicted to exhibit a maximum at intermediate molecular weights with capillarity-ductile failure occurring at low molecular weights and cohesive fracture, at high molecular weights. The results are compared to experiments on polyethylenes. There is general qualitative agreement especially with the behavior of low and high molecular weights where capillarity and fracture occur. The tendency to necking and ductile failure differs considerably among melts and is more pronounced in high-density than in low-density polyethylenes. The application to continuous spinline behavior is discussed, and draw resonance is suggested to be the continuous process analogue of ductile failure.