We simulate the spatial and temporal evolution of inhomogeneous flow fields in viscometric devices such as cylindrical Couette cells. The computations focus on a class of two species elastic network models which are prototypes for a model which can capture, in a self-consistent manner, the creation and destruction of elastically active network segments as well as diffusive coupling between the microstructural conformations and the local state of stress in regions with large spatial gradients of local deformation. For each of these models, the "flow curve" of stress and apparent shear rate resulting from an assumption of homogeneous deformation is nonmonotonic and linear stability analysis shows that the region of nonmonotonic response is unstable. Steady state calculations of the full inhomogeneous flow field lead to localized shear bands that grow linearly in extent across the gap as the apparent shear rate is incremented. Time-dependent calculations in step strain experiments and in start up of steady shear flow show that the velocity profile in the gap and the total stress measured at the bounding surfaces are coupled and evolve in a complex nonmonotonic manner as the shear bands develop. These spatio-temporal dynamics are consistent with time-resolved particle imaging velocimetry measurements in both concentrated solutions of monodisperse entangled polymers and in wormlike micellar solutions. The computational results have a number of implications for experimental observations of "apparent" or "gap-averaged" quantities in nearly viscometric devices, and lead to plateaus or "yield-like" transitions in the steady flow curve and deviations from the Lodge-Meissner relation in nonideal step shearing deformations. (C) 2008 The Society of Rheology.