|Abstract||This paper is a study of the class of groups that are not of finite exponent but all proper homomorphic images of these groups are of finite exponent. This class is denoted by X. Chapter II develops basic properties enjoyed by these groups as well as a wreath product method that allows the construction of groups in the class X. Of the basic properties developed in chapter II, the following three are used repeatedly in developing this paper. (1) If G is in the class X and A and B are nontrivial normal subgroups of G, then A and B intersect nontrivially. (2) If G is in the class X, then the Fitting subgroup of G is torsion free abelian. (3) If G is in the class X, is not periodic and if H is an ascendant subgroup of G, then H is not periodic. Chapter III is an in depth study of the Fitting subgroup of an X group. It is proved that if F is the Fitting subgroup of G with G in X, F not equal to 1, and if H is the Hirsch-Plotkin radical of G, then H=F. Another result of chapter III is that if G is in the class X and G is abelian, then G is infinite cyclic; if G is nilpotent then G is infinite cyclic; if G is supersolvable then G is either infinite cyclic or the infinite dihedral group; if G is metabelian then G is a finite extension of a free abelian group of finite rank. Chapter IV is a study of periodic X groups. Also given is an example of a just-infinite group which has not appeared in the literature. Some open questions on the class X have been pointed out throughout the paper.