The relation between mathematics and physics has a long history, in which the role of number theory and of other more abstract parts of mathematics has recently become more prominent. More than ten years after a first meeting in 1989 between number theorists and physicists at the Centre de Physique des Houches, a second 2-week event focused on the broader interface of number theory, geometry, and physics. This book is the result of that exciting meeting, and collects, in 2 volumes, extended versions of the?lecture courses, followed by?shorter texts on special topics, of eminent mathematicians and physicists. The present volume has three parts: Conformal Field Theories,?Discrete Groups, Renomalization. The companion volume is subtitled:?On Random Matrices, Zeta Functions and?Dynamical Systems?(Springer, 3-540-23189-7).