Two kinds of selective advantage have been suggested for sexual reproduction; that it accelerates evolutionary adaptation, and that by producing more variable progeny it increases the chance of producing some offspring of very high fitness.The rate of evolutionary adaptation of sexual and asexual populations is compared. If N is the population size, I the number of loci at which at any time favourable mutations are possible but have not yet occurred, u the mutation rate per locus and s the selective advantage per locus, sexual reproduction will not accelerate evolution if l/2 Nu < In 2ONs, or, very approximately, if N < 1jlOu. For larger populations sex does accelerate evolution, by a factor very approximately equal to 1.It is shown that the immediate advantage of more variable progeny arises only in the very extreme case in which the environment fluctuates unpredictably, so that correlations between different features of the environment commonly change sign between one generation and the next.It is suggested that the most important advantage of sex arises when two genetically different populations migrate into a new environment, in which the best adapted genotype is a combination of genes from the two invading populations.