Brownian dynamics simulations can be made more efficient by incorporating the idea of importance sampling. By introducing and compensating a bias in favor of those configurations which mainly contribute to the average of a given quantity of interest, one can considerably reduce the variance of the stochastic simulation results. This idea can be applied to general stochastic differential equations of motion and is hence not restricted to Brownian dynamics simulations. The construction of variance reduced simulations requires an approximate understanding of the dynamics described by the underlying stochastic differential equation. The basic procedure is first developed in general and then illustrated for the example of a Hookean dumbbell solution in start up of steady shear flow. Particular emphasis is put on the development of a clearly structured formulation of the procedure which immediately allows for variance reduced simulations of nonlinear models. Possible applications in polymer kinetic theory and in the flow calculation of viscoelastic liquids are discussed.