  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 {\it finitely generated abelian} groups and we
consider the problem of characterizing those groups which arise in some fixed
classes of rings $\mathcal C$, namely the integral domains, the torsion free
rings and the reduced rings. To determine the realizable groups we have to
establish what finite abelian groups $T$ (up to isomorphism) occur as torsion
subgroup of $A^*$ when $A$ varies in $\mathcal C$, and on the other hand, we
have to determine what are the possible values of the rank of $A^*$ when
$(A^*)_{tors}\cong T$. Most of the paper is devoted to the study of the class
of torsion-free rings, which needs a substantially deeper study.
