The reaction--diffusion process corresponding to the Fisher--Kolmogorov equation is studied by means of a discrete multivariate master equation. For travelling wave fronts the stability criterion necessary for the applicability of a system--size expansion is shown to be violated due to the existence of a zero mode of the first variational equation. This zero mode is connected to the translational invariance of the system. Performing stochastic simulations of the master equation in a wide range of parameters it is demonstrated that for finite size of the system (up to about 10 7 particles in the frontal region) a rather large fluctuation effect on the wave propagation speed results: In general, the asymptotic wave speed lies below the stable, minimal speed which is given by a theorem of Kolmogorov for the macroscopic equation. The wave front position exhibits a diffusion--type behaviour associated with translative fluctuations along the propagation direction. 1 Introduction The Fishe...