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. 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 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 This volume does not pretend to be comprehensive, nor is it a historical document.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 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.The Fields Medallists' Lectures is now available on -ROM. Sections can be accessed at the touch of a button, and similar topics grouped together using advanced keyword searches.Contents: (1936) L V AHLFORS:AutobiographyCommentary on: Zur Theorie der Überlagerungsflächen (1935)Quasiconformal Mappings, Teichmüller Spaces, and Kleinian Groups(1950) L SCHWARTZ:The Work of L Schwartz by H BohrBiographical NoticeCalcul Infinitesimal Stochastique(1958) K F ROTH: The Work of K F Roth by H DavenportBiographical NoticeRational Approximations to Algebraic Numbers(1958) R THOM:The Work of R Thom by H HopfAutobiography(1962) L HÖRMANDER:Hörmander's Work on Linear Differential Operators by L GårdingAutobiographyLooking forward from ICM 1962(1966) M F ATIYAH:L'oeuvre de Michael F Atiyah by H CartanBiographyThe Index of Elliptic Operators(1966) S SMALE:Sur les Travaux de Stephen Smale by R ThomBiographical NoticeA Survey of Some Recent Developments in Differential Topology(1970) A BAKER:The Work of Alan Baker by P TuránBiographyEffective Methods in the Theory of NumbersEffective Methods in Diophantine ProblemsEffective Methods in Diophantine Problems. II. CommentsEffective Methods in the Theory of Numbers/Diophantine Problems(1970) S NOVIKOV:The Work of Serge Novikov by M F AtiyahBiographyRôle of Integrable Models in the Development of Mathematics(1974) D MUMFORD:The Work of David Mumford by J TateAutobiographyPattern Theory: A Unifying Perspective(1978) G A MARGULIS:The Work of Gregori Aleksandrovitch Margulis by J TitsBiographical NoticeOppenheim Conjecture(1982) A CONNES:The Work of Alain Connes by H ArakiBiographical NoticeBrisure de Symétrie Spontanée et Géométrie du Point de vue Spectral(1982) W P THURSTON:The Work of W Thurston by C T C Wall(1986) S K DONALDSON:The Work of Simon Donaldson by M F AtiyahBiographical NoticeRemarks on Gauge Theory, Complex Geometry and 4-Manifold Topology(1986) M H FREEDMAN:The Work of M H Freedman by J MilnorBiographical NoticeBetti Number Estimates for Nilpotent Groups(1990) V F R JONES:The Work of Vaughan F R Jones by J S BirmanBiographical NoticeA Polynomial Invariant for Knots via von Neumann AlgebrasIndex for Subfactors(1990) S MORI:The Work of Shigefumi Mori by H HironakaBiographical NoticeBirational Classification of Algebraic Threefolds(1990) E WITTEN: The Work of E Witten by L D FaddeevThe Work of Edward Witten by M F AtiyahBiographical NoticeGeometry and Quantum Field Theory(1994) J BOURGAIN:The Work of Jean Bourgain by L CaffarelliBiographical NoticeHamiltonian Methods in Nonlinear Evolution Equations(1994) P L LIONS:The Work of Pierre-Louis Lions by S R S VaradhanBiographical NoticeOn Some Recent Methods for Nonlinear Partial Differential Equations(1994) J C YOCCOZ:Présentation de Jean-Christophe Yoccoz by A DouadyRecent Developments in Dynamics(1994) E I ZELMANOV:The Work of Efim Zelmanov by W FeitBiographical NoticeOn the Restricted Burnside ProblemReadership: Mathematicians and mathematical physicists.Key Features:Most of the articles are by well-known mathematiciansResults presented are important to analysis and numerical methodsContains an introduction to Jacques-Louis Lions