A necessary prerequisite of the scientific design of polymer processing operations is a fundamental understanding of the rheology of melts. In fiber spinning the dominant rheological response has simple extensional geometry and we require information about viscosity, elasticity, and fracture during such flows. Industry needs both absolute quantitative information and also inexpensive qualitative tests for multiple samples. Experimentation demands the greatest quantity of accurate information at minimum cost, and this philosophy underlies a critical revue of approaches to measuring the rheology of polymer melts under tension. The decisive techniques so far developed lack versatility and are not suitable for multiple testing. The continuous drawing of a monofilament, or laboratory spinning, is an excellent qualitative approach but sufficient experimental control has yet to be attained. Two indirect routes, through normal stresses to a “normal” viscosity and by steady shear/dynamic rheometry correlation to an “irrotational” viscosity, are superficially attractive but theoretical complexity is a major problem. The most promising industrial approach is the study of converging flows. Analysis of melt flowing from a reservoir into a die yields a simple method of obtaining apparent extensional rheology data which is compatible with, and supplementary to, the results offered by the direct methods of measurement. The analysis allows the ubiquitous capillary rheometer to become a simple extensional flow meter. It is the alliance of the definitive, but costly and temperamental, methods with inexpensive simple qualitative techniques that forges rheological tools for tackling industrial problems.