  We study the deep interplay between geometry of quadrics in d-dimensional
space and the dynamics of related integrable billiard systems. Various
generalizations of Poncelet theorem are reviewed. The corresponding analytic
conditions of Cayley's type are derived giving the full description of
periodical billiard trajectories; among other cases, we consider billiards in
arbitrary dimension d with the boundary consisting of arbitrary number k of
confocal quadrics. Several important examples are presented in full details
demonstrating the effectiveness of the obtained results. We give a thorough
analysis of classical ideas and results of Darboux and methodology of Lebesgue,
and prove their natural generalizations, obtaining new interesting properties
of pencils of quadrics. At the same time, we show essential connections between
these classical ideas and the modern algebro-geometric approach in the
integrable systems theory.
