Many well-studied coevolutionary interactions between predators and prey or hosts and parasites are mediated by quantitative traits. In some interactions, such as those between cuckoos and their hosts, interactions are mediated by the degree of phenotype matching among species, and a significant body of theory has been developed to predict the coevolutionary dynamics and outcomes of such interactions. In a large number of other cases, however, interactions are mediated by the extent to which the phenotype of one species exceeds that of the other. For these cases-which are arguably more numerous-few theoretical predictions exist for coevolutionary dynamics and outcomes. Here we develop and analyze mathematical models of interspecific interactions mediated by the extent to which the quantitative trait of one species exceeds that of the other. Our results identify important differences from previously studied models based on trait matching. First, our results show that cyclical dynamics are possible only if the strength of coevolutionary selection exceeds a threshold and stabilizing selection acts on the interacting traits. Second, our results demonstrate that significant levels of genetic polymorphism can be maintained only when cyclical dynamics occur. This result leads to the unexpected prediction that maintenance of genetic polymorphism is enhanced by strong selection. Finally, our results demonstrate that there is no a priori reason to expect the traits of interacting species should match in any literal sense, even in the absence of gene flow among populations.