The authors classify all groups all of whose subgroups of infinite index are Abelian. This class of groups is closed with respect to subgroups and quotient groups (but not a formation). Main theorem: If \(G\) is a non-Abelian group as defined, then \(G\) is finitely generated and \textit{either} (a) \(G/Z(G)\) is a just-infinite group with no Abelian subgroups of finite index and any two maximal Abelian subgroups of \(G/Z(G)\) have trivial intersection, \textit{or} (b) \(G\) is soluble with derived length at most \(3\), its largest periodic normal subgroup is finite Abelian, and the description is completed by giving the structure of six subclasses (not reproduced here). The proof rests on these alternatives: (i) \(G\) is not soluble-by-finite (leading to (a)), (ii) \(G\) is soluble-by-finite but not nilpotent-by-finite, (leading to the first four cases of (b)), and (iii) \(G\) is nilpotent-by-finite.