Statistical Mechanics is the study of systems where the number of interacting particles becomes infinite. In the last fifty years tremendous advances have been made which have required the invention of entirely new fields of mathematics such as quantum groups and affine Lie algebras. They have engendered remarkable discoveries concerning non-linear differential equations and algebraic geometry, and have produced profound insights in both condensed matter physics and quantum field theory. Unfortunately, none of these advances are taught in graduate courses in statistical mechanics. This book is an attempt to correct this problem. It begins with theorems on the existence (and lack) of order for crystals and magnets and with the theory of critical phenomena, and continues by presenting the methods and results of fifty years of analytic and computer computations of phase transitions. It concludes with an extensive presentation of four of the most important of exactly solved problems: the Ising, 8 vertex, hard hexagon and chiral Potts models.