Let \(\text{FC}^0\) be the class of all finite groups, and for each non-negative integer \(n\) let the class \(\text{FC}^{n+1}\) be defined by induction as the class of all groups \(G\) such that for every element \(x\in G\) the factor group \(G/C_G(\langle x\rangle^G)\) is in \(\text{FC}^n\). The \(\text{FC}^1\)-groups are precisely groups with finite conjugacy classes, and the class \(\text{FC}^n\) contains all finite groups and all nilpotent groups of class \(\leq n\). In the paper under review the authors take the theory of FC-groups as a model, and show that many properties of FC-groups have \(\text{FC}^n\)-analogues. We mention two of the easy to state consequences of one of the main results: Let \(G\in\text{FC}^n\). (i) The subgroup \(\gamma_{n+1}(G)\) is periodic and the elements of finite order in \(G\) form a subgroup of \(G\). (ii) If \(G\) is torsion free then \(G\) is nilpotent of class \(\leq n\). Some of the results obtained hold also for the larger class of \(\text{FC}^\infty\)-groups defined as the groups \(G\) with the property that for every \(x\in G\) the factor group \(G/C_G(\langle x\rangle^G)\) belongs to some \(\text{FC}^n\) (with \(n\) depending on \(x\)). In particular: Let \(G\in\text{FC}^\infty\). (i) The chief factors of \(G\) are finite and the maximal subgroups of \(G\) are of finite index in \(G\). (ii) If the normality is a transitive relation, then \(G\) is an FC-group.