This volume comprises the proceedings of the 1995 Cardiff symposium on Sieve Methods, Exponential sums, and their applications in number theory. Included are contributions from many leading international figures in this area which encompasses the main branches of analytic number theory. In particular, many of the papers reflect the interaction between the different fields of sieve theory, Dirichlet series (including the Riemann Zeta-function), and exponential sums, whilst displaying the subtle interplay between the additive and multiplicative aspects of the subjects. The fundamental problems discussed include recent work on Waring's problem, primes in arithmetical progressions, Goldbach numbers in short intervals, the ABC conjecture, and the moments of the Riemann Zeta-function.