The fourth Algorithmic Number Theory Symposium takes place at the U- versiteit Leiden, in the Netherlands, from July 2-7, 2000. Its organization is a joint e?ort of Dutch number theorists from Leiden, Groningen, Nijmegen, and Amsterdam. Sixinvitedtalksand36contributedtalksarescheduled. Thisvolumecontains the written versions of the talks, with the exception of two of the invited talks. Not included are: A rational approach to? by Frits Beukers (Utrecht) and The 40 trillionth binary digit of? is 0 by Peter Borwein (Burnaby, Canada). These talksare aimed at a wider audience, and formpart of thespecial ANTS IV event Pi in de Pieterskerk on July 5, 2000. This event includes an evening ceremony in which the tombstone of Ludolph van Ceulen is replaced. Van Ceulen, who was appointed to Leiden in 1600, calculated 35 decimals of?.Histombstonein the Pieterskerk, in which these decimals were engraved, disappeared in the 19th century. ANTS in Leiden is the fourth in a series of symposia that started in 1994. Previous locations were Cornell University, Ithaca, New York (1994), Universit´ e de Bordeaux I in Bordeaux, France (1996), and Reed College, Portland, Oregon (1998). The diversity of the papers contained in this volume shows that the maintheme of ANTS, algorithmicnumber theory, istaken ina broadsense. The number of submissions for the Leiden conference largely exceeded the physical limitationsof ourone-week schedule. Weare therefore con?dent thatwe areonly at the beginning of a continuing tradition.