The study of the structure of groups all of whose subgroups have a prescribed property has been a consistent theme in the theory of groups. One of the earliest examples of work in this area is the paper of Miller-Moreno [12J which gives a classification of all finite non-Abelian groups in which every proper subgroup is Abelian (such groups, without the finiteness assumption, are called MillerMoreno groups). The structure of finite non-nilpotent groups with all proper subgroups nilpotent is taken up by Schmidt [23] and later by Gol'fond [7]. In a somewhat different line, Baer [1] gives a complete description of Dedekind groups (=groups with all subgroups normal). Baer's results extend to infinite groups the classical theorem of Dedekind [5]. Various combinations of the above ideas have also been studied. For example, Romalis and Sesekin [20], [21] and [22], study rneta-Hamiltonian groups these are groups with all subgroups normal or Abelian. In the above papers, Romalis and Sesekin obtain information primarily about locally solvable meta-Hamiltonian groups. Recent results show that the structure of Miller-Moreno groups will, in general, by very complex. For example, Ol'ghankii [16] has recently constructed a torsion-free simple group in which every proper subgroup is cyclic; Ol'~hanki~ also constructs infinite periodic groups in which every proper subgroup has prime order. In a parallel development, Rips [A generalization of small cancellation theory] has constructed infinite 2-generator groups of prime exponent in which every proper subgroup is cyclic. These remarkable examples suggest that Miller-Moreno groups (and hence meta-Hamiltonian groups) will admit a structure theory only in the presence of some finiteness condition. As a consequence of work on minimal conditions, Phillips and Wilson [17] have determined, under a finiteness condition, those groups in which every