a natural choice of line tangent to a convex body at each point. One obtains these tangents by starting with the normals and using the complex structure to rotate 90 degrees. These distinguished tangent lines can be used to define the outer billiards map roughly as above. History Most people attribute the introduction of outer billiards to Bernhard Neumann in the late 1950s, though it seems that a few people cite an earlier construction in 1945, due to M. Day. Jürgen Moser popularized the system in the 1970s as a toy model for celestial mechanics. This system has been studied classically in the Euclidean plane, and more recently in the hyperbolic plane. One can also consider higher-dimensional spaces, though no serious study has yet been made. Bernhard Neumann informally posed the question as to whether or not one can have unbounded orbits in an outer billiards system, and Moser put it in writing in 1973. Sometimes this basic question has been called the Moser-Neumann question. This question, originally posed for shapes in the Euclidean plane and solved only recently, has been a guiding problem in the field. Moser-Neumann question Bounded orbits in the Euclidean plane In the 70's, Jürgen Moser sketched a proof, based on K.A.M. theory, that outer billiards relative to a 6-times-differentiable shape of positive curvature has all orbits bounded. In 1982, Raphael Douady gave the full proof of this result. A big advance in the polygonal case came over a period of several years when three teams of authors, Vivaldi-Shaidenko, Kolodziej, and Gutkin-Simanyi, each using different methods, showed that outer billiards relative to a quasirational polygon has all orbits bounded. The notion of quasirational is technical (see references) but it includes the class of regular polygons and convex rational polygons, namely those convex polygons whose vertices have rational coordinates. In the case of rational polygons, all the orbits are periodic. In 1995, Sergei Tabachnikov showed that outer billiards for the regular pentagon has some aperiodic orbits, thus clarifying the distinction between the dynamics in the rational and regular cases. In 1996, Philip Boyland showed that outer billiards relative to some shapes can have orbits which accumulate on the shape. In 2005, Daniel Genin showed that all orbits are bounded when the shape is a trapezoid, thus showing that quasirationality is not a necessary condition for the system to have all orbits bounded. (Not all trapezoids are quasirational.)