A surface diffusion model for filamentary crystal growth from the vapor phase is presented which accounts for extensive growth normal to the surface. The model is based on the mechanism by which adsorbed atoms are incorporated into surface steps and the anisotropy associated with the capture process. It is shown that filamentary crystals may be initiated at a variety of surface structures including dislocations or impurities but that neither of these is prerequisite. Rather, filamentary crystal growth depends on mobility differences of atoms adsorbed on various crystal planes and the mechanism of capture at surface steps. Results of computer calculations simulating filamentary crystal growth are presented.