**Sieve theory** is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, **sifted sets** of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit *X*. Correspondingly, the primordial example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms. In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be.