We determine the speed of a crystallization (or, more generally, a solidification) front as it advances into the uniform liquid phase after the system has been quenched into the crystalline region of the phase diagram. We calculate the front speed by assuming a dynamical density functional theory (DDFT) model for the system and applying a marginal stability criterion. Our results also apply to phase field crystal (PFC) models of solidification. As the solidification front advances into the unstable liquid phase, the density profile behind the advancing front develops density modulations and the wavelength of these modulations is a dynamically chosen quantity. For shallow quenches, the selected wavelength is precisely that of the crystalline phase and so well-ordered crystalline states are formed. However, when the system is deeply quenched, we find that this wavelength can be quite different from that of the crystal, so the solidification front naturally generates disorder in the system. Significant rearrangement and aging must subsequently occur for the system to form the regular well-ordered crystal that corresponds to the free energy minimum. Additional disorder is introduced whenever a front develops from random initial conditions. We illustrate these findings with simulation results obtained using the PFC model.