During development, most neurons become polarized when one neurite, generally the longest, becomes the axon and the other neurites become dendrites. The physical mechanism responsible for such length-related differentiation has not been established. Here, we present a model of neuronal polarization based upon the existence of a "determinant chemical' whose concentration at the neurite tips influences the growth rate of the neurite. Over an extended parameter range the equations describing the formation, transport, and consumption of this chemical and the resulting neurite growth undergo a winner-take-all instability, yielding rapid growth of one neurite (the axon) and diminished growth of all others. The behaviour of this model agrees well with the results of axotomy experiments and experiments in which growth-modulating substances are applied to individual growth cones. Possible candidates for the determinant chemical are discussed, and further experiments are proposed to test the model.