This book should be in the library of every working mathematician.

Many papers are ... broad in that they address the history of their subject and also make predictions about future developments. Many readers will be drawn to the general interest material ... Readers in search of controversy will find plenty ... especially intrigued by Arnold's elaborate schema in which the triple tetrahedron-octahedron-icosahedron corresponds to the triple reals-complexes-quaternions ... an excellent book.

This collection demonstrates well that mathematics is alive and vital.

You can read this book as if listening to a succession of high-powered old school friends who are passing through ... most tell you about the mathematics that animates them ... you get a good sense of what they do, what's difficult about it, and why it matters ... provocative remarks ... vigorous account of much of the Russian school of mathematics ... thought-provoking reflections about mathematical life and language ... The most pleasing feature of this handsome book is the emphasis on the unity of mathematics ... The connections many people here want to make between mathematics and physics, von Neumann algebras and knot theory, number theory and analysis, are not only fresh and vivid, but oddly coherent. They give a sense not only of mathematics undergoing one of its characteristic contractions around a few organising principles, but how productive this reorganisation can be.

One hundred years ago, David Hilbert's famous list of 26 mathematical problems began to catalyze the collective efforts of the world's mathematicians toward a century's worth of new research and achievement. The International Mathematical Union commissioned the current volume to do the same for the century just beginning. Fully half the contributors here own a Fields Medal, mathematics' highest honor (and that does not even count Andrew Wiles). Obviously, simply by dint of the prestige and caliber of the authors, this volume deserves reader attention and a place on every library's shelves. The essays themselves vary from the entirely technical to the purely personal. The sort of reading encounter they offer can set the direction of a whole career, so the undergraduate who picks up this volume now may expect to return here many times in the years to come.

The mere names of the editors ensure that [the book] is fascinating and exciting ... Several of the 29 authors touch on the nature and/or future of mathematics ... the most interesting essays are those whose authors get out on a limb and dogmatically announce, as saving truth, propositions radically different from common opinion ... Among the authors there are many famous pure mathematicians whose contributions constitute a smorgasbord of delicacies sufficient to satisfy every taste. Do sample them!

This collection of essays will reward and stimulate anyone who dips into it.It belongs on every mathematician's bookshelf and in the mathematics collection of every library.