Evolutionary game theory is a method of analysing the evolution of phenotypes when fitnesses are frequency-dependent. The assumption made about inheritance is the simplest possible one, that individuals produce offspring identical to themselves — i.e. parthenogenetic inheritance. Hence the method is not well suited for analysing the genetic structure of populations, or the way in which evolution depends on breeding systems. Essentially, it is concerned with deciding which phenotypes will win in competition in an evolving population. If the fitnesses of phenotypes are constant and independent of their frequencies, it is simply a matter of deciding which is the fittest; if this is difficult, optimisation methods may be useful. Game theory is relevant only when fitnesses vary with frequency. This paper presents a formal account of evolutionary game theory; applications to field and laboratory data are discussed by Maynard Smith (1979).