  This paper studies geometric and spectral properties of $S$-adic shifts and
their relation to continued fraction algorithms. These shifts are symbolic
dynamical systems obtained by iterating infinitely many substitutions. Pure
discrete spectrum for $S$-adic shifts and tiling properties of associated Rauzy
fractals are established under a generalized Pisot assumption together with a
geometric coincidence condition. These general results extend the scope of the
Pisot substitution conjecture to the $S$-adic framework. They are applied to
families of $S$-adic shifts generated by Arnoux-Rauzy as well as Brun
substitutions. It is shown that almost all of these shifts have pure discrete
spectrum. Using $S$-adic words related to Brun's continued fraction algorithm,
we exhibit bounded remainder sets and natural codings for almost all
translations on the two-dimensional torus. Due to the lack of self-similarity
properties present for substitutive systems we have to develop new proofs to
obtain our results in the $S$-adic setting.
