  We introduce a new class of billiard systems in the plane, with boundaries
formed by finitely many arcs of confocal conics such that they contain some
reflex angles. Fundamental dynamical, topological, geometric, and arithmetic
properties of such billiards are studied. The novelty, caused by reflex angles
on boundary, induces invariant leaves of higher genera and dynamical behaviour
different from Liouville-Arnold's theorem. Its analogue is derived from the
Maier theorem on measured foliations. A local version of Poncelet theorem is
formulated and necessary algebro-geometric conditions for periodicity are
presented. The connection with interval exchange transformation is established
together with Keane's type conditions for minimality. It is proved that the
dynamics depends on arithmetic of rotation numbers, but not on geometry of a
given confocal pencil of conics.
