We present a mathematical model for the polymerization and depolymerization of actin filaments incorporating the diffusion of actin monomers. For this model we observe polymerization fronts under conditions of spatially homogeneous regulation of actin dynamics. These are found analytically and their emergence is confirmed by simulations. We will discuss the properties of these fronts in dependence on polymerization dynamics and the diffusion coefficient for monomers. Moreover, some simulation results for inhomogeneous regulation of polymerization will illustrate the displacement of the cytoskeleton of a resting cell in the initiation of movement.