Among the modern methods used to study prime numbers, the sieve has been one of the most efficient. Originally conceived by Linnik in 1941, the large sieve has developed extensively since the 1960s, with a recent realization that the underlying principles were capable of applications going well beyond prime number theory. This book develops a general form of sieve inequality, and describes its varied applications, including the study of families of zeta functions of algebraic curves over finite fields; arithmetic properties of characteristic polynomials of random unimodular matrices;...