Magnetic domain walls are proposed as data carriers in future spintronic devices, whose reliability depends on a complete understanding of the domain wall motion. Applications based on an accurate positioning of domain walls are inevitably influenced by thermal fluctuations. In this letter, we present a micromagnetic study of the thermal effects on this motion. As spin-polarized currents are the most used driving mechanism for domain walls, we have included this in our analysis. Our results show that at finite temperatures, the domain wall velocity has a drift and diffusion component, which are in excellent agreement with the theoretical values obtained from a generalized 1D model. The drift and diffusion component are independent of each other in perfect nanowires, and the mean square displacement scales linearly with time and temperature.