Although the Fields medal does not have the same public recognition as the Nobel prizes, they share a similar intellectual standing. It is restricted to one field - that of mathematics - and an age limit of 40 has become an accepted tradition. Mathematics has in the main been interpreted as pure mathematics, and this is not so unreasonable since major contributions in some applied areas can (and have been) recognized with Nobel prizes. The restriction to 40 years is of marginal significance, since most mathematicians have made their mark long before this age. A list of Fields medallists and their contributions provide an overview of mathematics over the past 60 years. It highlights the areas in which, at various times, greatest progress has been made. This volume does not pretend to be comprehensive, nor is it an historical document. On the other hand, it presents contributions from 22 Fields medallists and so provides a highly interesting and varied picture. The contributions themselves represent the choice of the individual medallists. In some cases the articles relate directly to the work for which the Fields medals were awarded. In other cases new articles have been produced which relate to more current interests of the medallists. This indicates that while Fields medallists must be under 40 at the time of the award, their mathematical development goes well past this age. In fact the age limit of 40 was chosen so that young mathematicians would be encouraged in their future work.