Brownian dynamics simulations of a bead-spring chain are used to predict the shear and extensional flow properties of dilute polystyrene "Boger fluids" in the filament-stretching device.;Using an accurate representation of the inverse Langevin function for the elastic ＇＇spring" force of a freely jointed chain, and parameter values extracted from Zimm molecular theory combined with the molecular properties of the polystyrene solutions, quantitatively accurate predictions of the steady-state shear viscosity and first normal stress difference are obtained. The growth of the uniaxial extensional viscosity after startup of steady extension is also reasonably accurately predicted for solutions of 1.95 and 3.9 million molecular weight polystyrene, but for higher polystyrene molecular weights there is a growing overprediction of the high-strain plateau viscosity and a growing lag in the predicted stress growth relative to the measured one. Introduction of "deformation-dependent drag" into the simulations only worsens the predictions. In deformation histories in which there is a sudden step up in the extension rate or a re-start of extensional flow after a pause, the stress rises very rapidly in a viscous-like manner. The simulations show that this behavior occurs because the conformations of the subpopulation of molecules that contributes most to the stress contains long fully extended portions. (C) 2000 The Society of Rheology. [S0148-6055(00)00902-0].