Let R be a commutative ring and C a class of indecomposable R-modules. The Krull-Schmidt property holds for C if, whenever G1 ⊕ ··· ⊕ Gn H1 ⊕ ··· ⊕ Hm for Gi, Hj ∈ C, then n = m and, after reindexing, Gi Hi for all i ≤ n. The main purpose of this thesis is to investigate Krull-Schmidt properties of certain classes of modules over Non-Noetherian rings. Particularly weakly Matlis domains, strong Mori domains and Marot rings, all of which are among the class of Non-Noetherian rings, are studied. wweak isomorphism types are defined and the conditions when they coincide for torsionless modules over weakly Matlis domains are discussed. With the help of this comparison, the Krull-Schmidt property of w-ideals of a strong Mori domain is characterized. Also, the same property for overrings of a strong Mori domain is examined. Some useful results for a Marot ring with ascending condition on its regular ideals are obtained. Krull-Schmidt property on regular ideals of such a ring is studied and a characterization is given. Furthermore, the same property is discussed for overrings of a Marot ring.