We describe an algorithm for performing Brownian dynamics simulations of particles diffusing on S-3, a hypersphere in four dimensions. The system is chosen due to recent interest in doing computer simulations in a closed space where periodic boundary conditions can be avoided. We specifically address the question how to generate a random walk on the 3-sphere, starting from the solution of the corresponding diffusion equation, and we also discuss an efficient implementation based on controlled approximations. Since S-3 is a closed manifold (space), the average square displacement during a random walk is no longer proportional to the elapsed time, as in R-3. Instead, its time rate of change is continuously decreasing, and approaches zero as time becomes large. We show, however, that the effective diffusion coefficient can still be obtained from the time dependence of the square displacement. (C) 2003 American Institute of Physics.