The goal of the book is to present, in a complete and comprehensive way, areas of current research interlacing around the Poncelet porism: dynamics of integrable billiards, algebraic geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics. The most important results and ideas, classical as well as modern, connected to the Poncelet theorem are presented, together with a historical overview analyzing the classical ideas and their natural generalizations. Special attention is paid to the realization of the Griffiths and Harris programme about Poncelet-type problems and addition theorems. This programme, formulated three decades ago, is aimed to understanding the higher-dimensional analogues of Poncelet problems and the realization of the synthetic approach of higher genus addition theorems.

Introduction to Poncelet Porisms.- Billiards First Examples.- Hyper-Elliptic Curves and Their Jacobians.- Projective geometry.- Poncelet Theorem and Cayleys Condition.- PonceletDarboux Curves and SiebeckMarden Theorem.- Ellipsoidal Billiards and their Periodical Trajectories.- Billiard Law and Hyper-Elliptic Curves.- Poncelet Theorem and Continued Fractions.- Quantum Yang-Baxter equation and (2-2)-correspondences.- Bibliography.- Index.