Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems

This is an updated and expanded version of our earlier survey article \cite{Gut5}. Section $\mathrm{\S }1$ introduces the subject matter. Sections $\S 2 - \S 4$ expose the basic material following the paradigm of elliptic, hyperbolic and parabolic billiard dynamics. In section $\mathrm{\S }5$ we report on the recent work pertaining to the problems and conjectures exposed in the survey \cite{Gut5}. Besides, in section $\mathrm{\S }5$ we formulate a few additional problems and conjectures. The bibliography has been updated and considerably expanded.