In rheology the use of the concepts of elasticity—or mechanical stored energy—or recoverable shear in steady flow has been used for some time, but by no means universally accepted. In this paper, the foundations of the statistical mechanics of polymer solutions, as developed by Kuhn, Rouse, and Zimm, together with numerous others, are reviewed for the purpose of developing the basic ideas of elasticity and birefringence in steady flow and investigating what assumptions and limitations the present theory has. The quantitative relations between a number of mathematical theories are tabulated and a survey of the experimental results is given. The questions still unsolved are formulated. Especially the correlation among viscosity, flow birefringence, dynamic properties, and normal stress measurements is emphasized.