For a given finite group \(G\), a ring \(R\) is said to be \(G\)-adapted if \(R\) is an integral domain of characteristic \(0\) in which no prime divisor of \(|G|\) is invertible. Three general results are established concerning units in group rings. The first result gives a characterization of finite groups \(G\) and \(G\)-adapted rings \(R\) such that the group ring \(RG\) has only trivial units. This generalizes the classical result by G. Higman for integral group rings \(\mathbb{Z} G\) with finite \(G\). Next the authors give a formula for the rank of the central units of \(\mathbb{Z} G\) with finite \(G\). This was independently obtained also by \textit{R. A. Ferraz} [in J. Algebra 279, No. 1, 191-203 (2004; Zbl 1080.16025)]. Finally, the authors' third main result says that if \(G\) is finite and \(R\) is a \(G\)-adapted ring such that the unit group of \(R\otimes_\mathbb{Z}\mathbb{Z}[\zeta_{|G|}]/|G|\) is torsion, then the central units in \(RG\) are all trivial if and only if the unit group of \(R\otimes_\mathbb{Z}\mathbb{Z}[\chi]\) is torsion for all complex characters \(\chi\) of \(G\).