This book explores the foundations of hamiltonian dynamical systems and statistical mechanics, in particular phase transitionsc, from the point of view of geometry and topology. A broad participation of topology in these fields has been lacking and this book will provide a welcome overview of the current research in the area, in which the author himself is a pioneer. Using geometrical thinking to solve fundamental problems in these areas, compared to the purely analytical methods usually used in physics could be highly productive. The author skillfully guides the reader, whether mathematician or physicists through the background needed to understand and use these techniques.Dr. Marco Pettini is affiliated with the Istituto Nazionale di Astrofisica Ãoâe┐" Osservatorio Astrofisico di Arcetri in Firenze, Italy.