The nonlinear transient extensional and steady-state shear rheological properties of dilute polystyrene solutions in a theta solvent are predicted using Brownian Dynamics (BD) simulations with the bead-spring model. Full hydrodynamic interaction is incorporated into BD simulations using the Rotne-Prager tensor. The hydrodynamic interaction parameter h* is obtained a priori by matching the drag force from a fully extended bead-spring model in extensional flow with that from Batchelor's theory for a cylindrical rod. The agreement between experimental data [Gupta et al. (2000)] and simulation results for the transient Trouton ratio vs. strain is good from low to medium strains. However, the plateaus at high strains predicted by the simulations are higher than measured. A coil-stretch hysteresis is predicted the first time for a synthesized polymer and we find that the threshold value of the ratio of effective drag coefficients in the stretched vs. coiled states, sigmastretch/sigmacoil is about 4.5 for the occurrence of hysteresis, close to the value observed in experiments for DNA. In steady shear flow, the simulated first normal stress coefficient does not quantitatively match the experimental results, in part because the number of beads N required to represent the hydrodynamic interactions accurately is much higher than we can afford to use. However, a negative second normal stress coefficient is predicted. Bead-spring Brownian dynamics simulations are then used to model flow-induced chain scission in dilute polymer solutions in a planar cross-slot flow. In the simulations, we observed both chain halving and quartering as reported in the experimental literature. However, in contrast to the common view that breakage occurs only in the stagnation point region, we find that the strong shearing flow generated near the walls of the inlet channel pre-stretches polymer molecules considerably, leading to breakage near the corner where an extensional flow is present. We also predict flow birefringence, which supports our finding that shear plays a significant role in chain scission in this geometry. Our findings show that combining hydrodynamic interactions or complex fluid fields with realistic Brownian dynamics simulations of bead-spring models can now shed light on previously intractable issues in polymer fluid mechanics and rheology.