  A Sturmian sequence is an infinite nonperiodic string over two letters with
minimal subword complexity. In two papers, the first written by Morse and
Hedlund in 1940 and the second by Coven and Hedlund in 1973, a surprising
correspondence was established between Sturmian sequences on one side and
rotations by an irrational number on the unit circle on the other. In 1991
Arnoux and Rauzy observed that an induction process (invented by Rauzy in the
late 1970s), related with the classical continued fraction algorithm, can be
used to give a very elegant proof of this correspondence. This process, known
as the Rauzy induction, extends naturally to interval exchange transformations
(this is the setting in which it was first formalized). It has been conjectured
since the early 1990s that these correspondences carry over to rotations on
higher dimensional tori, generalized continued fraction algorithms, and
so-called $S$-adic sequences generated by substitutions. The idea of working
towards such a generalization is known as Rauzy's program. Recently Berth\'e,
Steiner, and Thuswaldner made some progress on Rauzy's program and were indeed
able to set up the conjectured generalization of the above correspondences.
Using a generalization of Rauzy's induction process in which generalized
continued fraction algorithms show up, they proved that under certain natural
conditions an $S$-adic sequence gives rise to a dynamical system which is
measurably conjugate to a rotation on a higher dimensional torus. Moreover,
they established a metric theory which shows that counterexamples like the one
constructed in 2000 by Cassaigne, Ferenczi, and Zamboni are rare. It is the aim
of the present chapter to survey all these ideas and results.
