Interval exchange transformations are invertible, piecewise order preserving isometries of the unit interval with finitely many discontinuities. Starting from rotations of the circle, which they generalize, this talk will present their connections to flows on flat surfaces, rational billiards and symbolic coding. Recent results on diophantine approximation for interval exchange transformations will be presented. This talk will discuss joint work with M.
Boshernitzan, Y. Cheung and H. Masur, and D. Constantine.