The book serves two purposes.  The stated goal is to allow a prospective grad student in mathematics or a closely related subject to easily examine subject matter that will be regarded as core material at the graduate level, but might have been missed at the undergraduate level.  A second purpose the book could serve is to provide a list of advanced mathematical topics that might have broad value beyond that of progressing toward a degree.  The book's approach serves both of these goals very well.  Twenty discrete topics are reviewed separately, each of which provide definitions, theorems, examples, and problems that cover the core concepts.  This gives the reader a very clear idea as to whether the subject has been mastered, or if it demands a brush-up review or more substantial study to get up to speed.  The casual rather than formal style helps with this task, and highlights the fact that this isn't a mathematics textbook.  If it turns out the reader is weak on a topic, each chapter ends with a list of recommended books with indications of the contents, depth and style of coverage that each book offers.<br><br>Having read the book, and being familiar with about half a dozen of the topics covered, I consider the approach very effective and the material to be well chosen.  If there are any less important topics included, I would guess that these would be the four chapters added to the second edition, all of which involve number theory.  I've also found the casual style beneficial for looking up mathematical concepts.  A simple web search yields strict technical definitions and rough illustrative analogies of terms and concepts in advanced mathematics, but it's harder to find a quick, well-informed summary clarifying topical questions.<br><br>I am not a mathematics graduate student, but I found the book invaluable, and I'm convinced I would have benefitted from having this book as a freshman in college, or even as a junior in high school.  It's not a mathematics textbook, it's a list of places to direct one's attention in the field of advanced mathematics.