There exist in certain nervous systems topological mappings of one set of spatially ordered nerve cells onto another, whose formation has not yet been satisfactorily explained. In this paper the role that competition may play in the development of such spatially patterned nervous connexions is discussed, and simple models incorporating competition and graded affinity between axons and postsynaptic sites have been tested by computer simulation. It is shown that the conditions for map making by this mechanism are that there is not only competition between axons for postsynaptic sites but also between postsynaptic sites for axons; and that this can be effected by imposing saturation conditions limiting the number of postsynaptic sites an axon may contact simultaneously and similarly the number of axon branches that can contact a postsynaptic cell. The possibility was investigated that competition models might lead to spreading or compression of the pattern of connexions in spatial mismatch experiments, such as have been done on the visual system of lower vertebrates. Two classes of these experiments exist: (a) those in which no size disparity is present between the set of presynaptic branches and the set of postsynaptic sites (for example, the Xenopus compound eye experiments); competition models predict an obligatory spreading of connexions without postulating regulation; (b) those in which a size disparity does exist (for example, a complete retina being made to regenerate into a half tectum, or vice versa); competition models predict spreading-compression only if the number of possible presynaptic branches available for contact formation is once more made equal to the number of available sites. The interpretation of other experimental designs is discussed. The importance of the establishment of transient or temporary contacts during development is emphasized.