  We consider Arnoux-Rauzy subshifts $X$ and study various combinatorial
questions: When is $X$ linearly recurrent? What is the maximal power occurring
in $X$? What is the number of palindromes of a given length occurring in $X$?
We present applications of our combinatorial results to the spectral theory of
discrete one-dimensional Schr\"odinger operators with potentials given by
Arnoux-Rauzy sequences.
