Poncelet's theorem is one of the most beautiful and important results of projective geometry. In this book, for the first time in world literature, Poncelet-type theorems are systematically described, as well as their natural more multidimensional generalizations and applications in the field of mechanics and geometry. The main goal of this book is to create and implement a program for a synthetic approach to addition theorems in higher genera. The implementation of this program is to study the far-reaching connections between the dynamics of integrable billiards and the geometry of quadric and hyperelliptic Jacobians.