People like to say that the Fields Medal is “the equivalent of a Nobel Prize in mathematics.” I guess the reason for that is simply that there is no Nobel Prize for mathematics, so we latch on to a prestigious award and call it that. But in fact, the Fields Medal is very different: it is a prize for young mathematicians who have made a significant impact on the field, intended to encourage them to continue doing so. Nobel Prizes tend, instead, to honor long and successful careers even though they usually name some specific breakthrough (usually achieved many years before) as the basis for the prize. The Wolf Prize in Mathematics and the Abel Prize, both of which focus on a lifetime of achievement, are much closer to being “mathematical equivalents” of the Nobel.

While there is no formal rule that Fields Medalists must be under 40, in practice that rule has been followed consistently. (It is hard to imagine the Nobel committees doing that.) All of the winners have been brilliant young mathematicians, but it is arguable that not all of them fulfilled their promise quite as gloriously as others. Still, any list of significant twentieth century mathematicians will include almost all of them.

The first Fields Medals were awarded in 1936 at the International Congress of Mathematicians in Oslo. When the ICM resumed after the war, in 1950, the medals began to be awarded every four years. Each time there have always been at least two winners, but the more common number is four.

The tradition is for a senior mathematician to give a short lecture at ICM on the work of the new medalist. This volume collects these lectures (as published in the proceedings of each congress), a short biographical essay on each medalist, and a mathematical paper chosen by the winner. Most are reprints, but some seem to have been written for this volume. The title, then, is not quite correct: most of the articles included are *not* transcripts of lectures by the winners of Fields Medals. On the other hand, allowing the winners to choose how they would be represented does give them a chance to include something broader, or something they are interested in now, rather than what they said way back when.

The first edition of *Fields Medallists’ Lectures *covered the period between 1936 and 1994; this second edition includes the medals from 1998 and 2002. The editors tell us that “this volume does not pretend to be comprehensive,” and indeed many winners are omitted, including such household names as Atle Selberg, Jean-Pierre Serre (an Abel Prize winner), and Alexandre Grothendieck. Presumably they just weren’t interested in being included.

While this is not quite the reference volume I was expecting, what is here is quite interesting. I am happy to have the brief biographies and the photographs that come with them, and many of the included articles are quite useful surveys of their authors’ work.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and the editor of MAA Reviews.