Finitely generated abelian groups of units

In 1960, Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In this paper, we address Fuchs' question for finitely generated abelian groups and consider the problem of characterizing those groups which arise in some fixed classes of rings C, namely, the integral domains, the torsion free rings and the reduced rings. Most of the paper is devoted to the study of the class of torsion-free rings, which needs a substantially deeper study.