Absolutely fascinating from beginning to end. Ian Stewart, New Scientist This is a book rich in ideas. They seem to burst forth from almost every page ... I suspect that the author's hope that 'even the experts will find some new enlightenment here' will be realized. The Mathematics Teacher A very original book ... The lectures are self-contained for the graduate reader ... full of new ideas, and thus of real interest also to experts. Zentralblatt für Mathematik John Horton Conway's unique approach to quadratic forms was the subject of the Hedrick Lectures given by him in August of 1991 at the Joint Meetings of the Mathematical Association of America and the American Mathematical Society in Orono, Maine. This book presents the substance of those lectures. The book should not be thought of as a serious textbook on the theory of quadratic forms - it consists rather of a number of essays on particular aspects of quadratic forms that have interested the author. The lectures are self-contained and will be accessible to the generally informed reader who has no particular background in quadratic form theory. The minor exceptions should not interrupt the flow of ideas. The Afterthoughts to the Lectures contain discussion of related matters that occasionally presuppose greater knowledge.