  In \cite[Problem 72]{Fuchs60} 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 a previous paper \cite{DDcharp} we dealt with finite characteristic
rings. In this paper we consider Fuchs' question for finite groups and we
address this problem in two cases. Firstly, we study the case of torson-free
rings and we obtain a complete classification of the finite groups of units
which arise in this case. Secondly, we examine the case of characteristic zero
rings obtaining, a pretty good description of the possible groups of units
equipped with families of examples of both realizable and non-realizable
groups. The main tools to deal with this general case are the Pearson and
Schneider splitting of a ring \cite{PearsonSchneider70}, our previous results
on finite characteristic rings \cite{DDcharp} and our classification of the
groups of units of torsion-free rings.
  As a consequence of our results we completely answer Ditor's question
\cite{ditor} on the possible cardinalities of the group of units of a ring.
