Between the 17th and the 21st of July, 1995 a group of about fifty mathematicians gathered in Cardiff for the Symposium on Sieve Methods, Exponential Sums and their Applications in Number Theory. They exchanged lectures and informal discussions and posed problems in the broad subject area defined by the title of the Symposium. This volume of papers gives an account of work described at the Symposium. Certain articles include a description of work done after the Symposium closed, some of this being prompted by questions posed during the Symposium.

Many of the contributions involve connections between the additive and multiplicative sides of analytic number theory, an interplay which the Symposium had been planned both to reflect and to promote. The reader will find represented here most of the branches of this subject which, as this volume demonstrates, has its roots in antiquity with the Sieve of Eratosthenes, yet is currently vibrant and receiving fresh stimulus from such diverse topics as trace formulae and elliptic curves.

Many of the participants at the Symposium were supported by the Engineering and Physical Sciences Research Council of the U.K. Others were able to bring their own support with them. The administrative costs were underwritten by the London Mathematical Society. The organisers, who are also the editors of this volume, are grateful to all the sources of support which made the Symposium possible.

The papers in this volume have been refereed to the standards required by leading journals; we take this opportunity to thank the referees for their work.