More than 50 years ago, Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. After introducing the problem and known results in the first part of the talk, I will present an overview of my joint work with Keir Lockridge on this problem. We answered this question for various classes of groups including indecomposable abelian groups, dihedral groups, quaternion groups, and some $p$-groups. This work also gave us several new characterizations of Mersenne primes and Fermat primes.