  In this paper we undertake the general study of the Abelian complexity of an
infinite word on a finite alphabet. We investigate both similarities and
differences between the Abelian complexity and the usual subword complexity.
While the Thue-Morse minimal subshift is neither characterized by its Abelian
complexity nor by its subword complexity alone, we show that the subshift is
completely characterized by the two complexity functions together. We give an
affirmative answer to an old question of G. Rauzy by exhibiting a class of
words whose Abelian complexity is everywhere equal to 3. We also investigate
links between Abelian complexity and the existence of Abelian powers. Using van
der Waerden's Theorem, we show that any minimal subshift having bounded Abelian
complexity contains Abelian k-powers for every positive integer k. In the case
of Sturmian words we prove something stronger: For every Sturmian word w and
positive integer k, each sufficiently long factor of w begins in an Abelian
k-power.
