  Return words constitute a powerful tool for studying symbolic dynamical
systems. They may be regarded as a discrete analogue of the first return map in
dynamical systems. In this paper we investigate two abelian variants of the
notion of return word, each of them gives rise to a new characterization of
Sturmian words. We prove that a recurrent infinite word is Sturmian if and only
if each of its factors has two or three abelian (or semi-abelian) returns. We
study the structure of abelian returns in Sturmian words and give a
characterization of those factors having exactly two abelian returns. Finally
we discuss connections between abelian returns and periodicity in words.
