Originally, a billiard is a dynamical system describing the motion of a point particle in a connected, compact domain Q of some d-dimensional torus. Inside Q the motion is uniform, while at the boundary of Q (the "scatterers"), the particle is reflected elastically (or, just as light is reflected in geometrical optics).
This notion of a billiard also covers the case of several elastically colliding spheres, since these can be treated like a single particle moving in many dimensions, among different obstacles (scatterers).

However, the word 'billiard' today is used in a much wider sense. We just mention a few possible generalizations, to illustrate the difficulty of giving a general definition:

• infinitely many particles
• variable number of particles due to the presence of a particle reservoir
• soft scatterers with some potential, where reflection is not momentary, but the particle(s) may enter the scatterer to some extent
• presence of a force field spoiling the uniformity of motion between two reflections
• boundary conditions representing a thermal bath (and/or particle reservoir)
• moving/rotating scatterers

We could say that any dynamical system, where motion can be viewed as a series of collisions and 'interaction-free' flight between two collisions, is a billiard. Due to the wide range of systems covered, we do not attempt to give a general definition of a billiard. Instead, we agree that the word billiard by itself will always mean the basic case, as defined here:

Some basic conventions: conventions.dvi or conventions.ps

Other notions of a billiard are indicated by adding distinctive phrases like `with force field', `thermostated', etc. We ask every contributor to exactly describe the setting, and fix the notation for their problem, if different from the one fixed above.