Unbounded orbits in the Euclidean plane In 2007, Richard Schwartz showed that outer billiards has some unbounded orbits when defined relative to the Penrose Kite, thus answering the original Moser-Neumann question in the affirmative. The Penrose kite is the convex quadrilateral from the kites-and-darts Penrose tilings. Subsequently, Schwartz showed that outer billiards has unbounded orbits when defined relative to any irrational kite. An irrational kite is a quadrilateral with the following property: One of the diagonals of the quadrilateral divides the region into two triangles of equal area and the other diagonal divides the region into two triangles whose areas are not rational multiples of each other. In 2008, Dmitry Dolgopyat and Bassam Fayad showed that outer billiards defined relative to the semidisk has unbounded orbits. The semidisk is the region one gets by cutting a disk in half. The proof of Dolgopyat-Fayad is robust, and also works for regions obtained by cutting a disk nearly in half, when the word nearly is suitably interpreted. Unbounded orbits in the hyperbolic plane In 2003, Filiz Doǧru and Sergei Tabachnikov showed that all orbits are unbounded for a certain class of convex polygons in the hyperbolic plane. The authors call such polygons large. (See the reference for the definition.) Filiz Doǧru and Samuel Otten then extended this work in 2011 by specifying the conditions under which a regular polygonal table in the hyperbolic plane have all orbits unbounded, that is, are large. Existence of periodic orbits In ordinary polygonal billiards, the existence of periodic orbits is a major unsolved problem. For instance, it is unknown if every triangular shaped table has a periodic billiard path. More progress has been made for outer billiards, though the situation is far from well understood. As mentioned above, all the orbits are periodic when the system is defined relative to a convex rational polygon in the Euclidean plane. Moreover, it is a recent theorem of Chris Culter (written up by Sergei Tabachnikov) that outer billiards relative to any convex polygon has periodic orbits—in fact a periodic orbit outside of any given bounded region. Open questions Outer billiards is a subject still in its beginning phase. Most problems are still unsolved. Here are some open problems in the area. Show that outer billiards relative to almost every convex polygon has unbounded orbits. Show that outer billiards relative to a regular polygon has almost every orbit periodic. The