The present paper explores the possible functional significance of the parallel orientation of parallel fibers in teleostean cerebellar and cerebelloid molecular layers, taking advantage of the restricted width of these molecular layers compared with mammalian ones and several specific configurations of granule cells. These configurations include: (i) a unilateral location, i.e. at only one (lateral) side of the molecular layer, giving rise to parallel fibers without bifurcation in a unidirectional molecular layer, where all parallel fibers conduct signals in the same direction; (ii) a bilateral location at both sides of the molecular layer giving rise to a bidirectional molecular layer where parallel fibers conduct signals in two opposite directions originating from two discrete sources; and (iii) a basal (or sometimes apical) location underneath (or opposite to) the layer of Purkinje cells, giving rise to a bidirectional molecular layer where parallel fibers conduct signals in two opposite directions originating from a continuous range of sources. It is argued that molecular layers with a bilateral location of granule cells, exemplified by the mormyrid lobus transitorius, represent an optimal configuration for the analysis of small temporal differences (up to 4 ms) between inputs to the right and left granule cell mass, by means of detection of the site of coincidence of parallel fiber activity running from left to right and vice versa. Morphological aspects that probably optimize such a function include not only the parallel course and bilateral origin of parallel fibers, but also their small diameter, large number and co-extensive location, as well as the sagittal orientation and the presence of many spines of Purkinje cell dendrites and the presence of stellate and other inhibitory interneurons. The only assumption underlying the present coincidence detection hypothesis is that Purkinje cells are supposed to be maximally stimulated by parallel fiber input when all spines are activated in such a way that their excitatory postsynaptic potentials reach the axon hillock simultaneously. For molecular layers with a unilateral location of granule cells, exemplified by the teleostean torus longitudinalis-tectal marginal parallel fiber system, a similar coincidence detecting mechanism is proposed on the basis of the presence of two populations of parallel fibers with slightly different conduction velocities. Such a system might be suitable to adapt the location of coincidence peaks to topographic maps present in deeper layers of nervous tissue. Molecular layers with basally (or apically) located granule cells as encountered in the teleostean corpus cerebelli, are probably involved in the analysis of specific spatio-temporal input waves directed centripetally towards different Purkinje cells.(ABSTRACT TRUNCATED AT 400 WORDS)