Piecewise isometries appear in applications ranging through digital dataprocessing (Sigma-Delta modulators), polygonal billiards and queueing theory. Their dynamics is usually a mixture of (quasi)periodic and chaotic motion, where it should be noted that the chaos is due to the discontinuities in the system, rather than to positive Lyapunov exponents. Except for a few special cases, which use number-theoretic peculiarities, the behaviour of piecewise isometries poses many unanswered questions. Already in dimension 1, there are piecewise isometries with very interesting properties, e.g. they can possess attractors of a multifractal nature, and carry multiple ergodic invariant measures.