Transient elasticity is a systematic generalization of viscoelasticity. Its purpose is to give a coherent description of non-Newtonian effects displayed by soft-matter systems, especially polymer melts and solutions. Using the concept of transient elasticity we describe here a hydrodynamic model for polymeric fluids, which is applicable for large amplitude deformations. We present an energy density with only two independent parameters, which is compatible with all thermodynamic requirements and which reduces for small deformations to models studied previously. The expression discussed is simple enough to allow full analytic treatment and shows semiquantitative agreement with experimental data. This model is used to capture many of the interesting effects thought to be characteristic of polymer rheology for large deformations including viscosity overshoot near the onset of shear flow, the onset of elongational flows in situations for which there is no stationary solution as well as shear thinning and normal stress differences for a large range of shear rates. In addition, we analyze how well our model accounts for empirical relations including the Cox-Merz rule, the Yamamoto relation, and Gleißle's mirror relations.