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 be (and have been) recognized with Nobel Prizes. A list of Fields Medallists and their contributions provides a bird's-eye view 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 This volume does not pretend to be comprehensive, nor is it a historical document. On the other hand, it presents contributions from Fields Medallists and so strives to provide an interesting and varied picture. This second edition of "Fields Medallists' Lectures" features additional contributions from the following Medallists: Kunihiko Kodaira (1954), Richard E. Borcherds (1998), William T. Gomers (1998), Maxim Knotsevich (1998), Curtis T. McMullen (1998) and Vladimir Voevodsky (2002).