In this work, we assess the capability of two thermodynamically consistent, microstructure-based models to predict the rheology and microstructure of a model wormlike micellar surfactant solution known to exhibit shear banding undergoing transient simple shear and large amplitude oscillatory shear deformations in a cylindrical Couette rheometer. The microstructure of the surfactant solution was simultaneously measured during the rheometric tests using small angle neutron scattering and this data is used along with the bulk rheometry to critically evaluate the models. The first model is a new, two-species model specifically developed for wormlike micelles, whereas the second model is the well-known Giesekus model with solvent and diffusion. The new model accounts for the microstructural dynamics including flow-induced micellar breakage and recombination, which is shown to be necessary to predict some of the measured rheology and microstructure. More species should be included to improve the quantitative comparison with experiments, as well as nonlinear relaxation, as judged by the success of the Giesekus model to represent certain features of the time-dependent data. More experimental research is needed to validate the role of flow-induced breakage in the nonlinear rheology of wormlike micellar surfactant solutions. (C) 2016 The Society of Rheology.