An adaptation in one lineage (e.g. predators) may change the selection pressure on another lineage (e.g. prey), giving rise to a counter-adaptation. If this occurs reciprocally, an unstable runaway escalation or 'arms race' may result. We discuss various factors which might give one side an advantage in an arms race. For example, a lineage under strong selection may out-evolve a weakly selected one ('the life-dinner principle'). We then classify arms races in two independent ways. They may be symmetric or asymmetric, and they may be interspecific or intraspecific. Our example of an asymmetric interspecific arms race is that between brood parasites and their hosts. The arms race concept may help to reduce the mystery of why cuckoo hosts are so good at detecting cuckoo eggs, but so bad at detecting cuckoo nestlings. The evolutionary contest between queen and worker ants over relative parental investment is a good example of an intraspecific asymmetric arms race. Such cases raise special problems because the participants share the same gene pool. Interspecific symmetric arms races are unlikely to be important, because competitors tend to diverge rather than escalate competitive adaptations. Intraspecific symmetric arms races, exemplified by adaptations for male-male competition, may underlie Cope's Rule and even the extinction of lineages. Finally we consider ways in which arms races can end. One lineage may drive the other to extinction; one may reach an optimum, thereby preventing the other from doing so; a particularly interesting possibility, exemplified by flower-bee coevolution, is that both sides may reach a mutual local optimum; lastly, arms races may have no stable end but may cycle continuously. We do not wish necessarily to suggest that all, or even most, evolutionary change results from arms races, but we do suggest that the arms race concept may help to resolve three long-standing questions in evolutionary theory. 'Wonderful and admirable as most instincts are, yet they cannot be considered as absolutely perfect: there is a constant struggle going on throughout nature between the instinct of the one to escape its enemy and of the other to secure its prey' (Charles Darwin, in Romanes I883).