A model for the coevolution of body size of predators and their prey is described. Body sizes are assumed to affect the interactions between individuals, and the Lotka-Volterra population dynamics arising from these interactions provide the driving force for evolutionary change. The space of phenotypes of predator and prey contains a region, oval in shape, in which the predator and prey species coexist. Within this region, evolutionarily stable strategies (ESSS) and evolutionary saddles may be found, and coevolution may tend to an ESS, develop a Red Queen dynamic, or move to predator extinction. Ten qualitatively distinct kinds of phenotype space are described, depending mainly on the number of ESSS and evolutionary saddles. These varied outcomes are in part due to the range of ways in which density-dependent selection within the prey interacts with density- and frequency-dependent selection on the prey due to the predator. The results point to a `loser wins' principle, in which the evolution leads to a weakening of the interaction between predator and prey. The results also illustrate the deterioration of the environment associated with each evolutionary step of the species and the lack of a net improvement in their mean fitness.