This volume consists of twenty survey articles on various topics in algorithmic number theory written by leading experts in the field. It starts with two introductory papers, one on the Pell equation (H. W. Lenstra, Jr.) and the other on basic algorithms in number theory (J. Buhler and S. Wagon). They are followed by eight articles covering the core of the field, including smooth numbers and the quadratic sieve (C. Pomerance), the number field sieve (P. Stevenhagen), primality testing algorithms (R. Schoof), lattices (H. W. Lenstra, Jr.), elliptic curves (B. Poonen), the arithmetic of number rings (P. Stevenhagen), computational number theory (A. Granville) and fast multiplication (D. Bernstein). The remaining ten articles contain surveys of specific topics, including discrete logarithms, cryptography, Arakelov class groups, computational class field theory and the algorithmic theory of zeta functions over finite fields. The book can be warmly recommended to anyone interested in the fascinating area of computational number theory.