Integrable billiards and rigidity

First I shall discuss various models of mathematical billiards and the notion of integrability. Next I shall focus on the so called Birkhoff-Poritsky conjecture, that the only Integrable billiards are circles and ellipses. I shall present several positive results (joint with A. E. Mironov and D. Tsodikovich) for Birkhoff and other type of billiards. If time permits, applications to spectral rigidity will be discussed.