In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars lie in arbitrary ring.The notion of purity was first introduced in abelian groups. This notion was extended to modules over principal ideal domains (PID). In 1952, Kaplansky introduced the notion of purity for modules over Dedekind domains. He has generalized most of the results in abelian groups to modules over a Dedekind domain. P.M.Cohn was first to define the purity in terms of tensor product for modules over an arbitrary ring and has shown that this notion is equivalent to one introduced for modules over a PID. Later Stenstrom has studied various notions of purity. Warfield Jr.has given projective characterization for purity and has introduced the notion of RD-purity which is same as purity in abelian groups. Hiremath has dualized the notion of purity as copurity and has made detailed investigation of copurity. In this thesis, we have made detailed study of cyclic purity and its dual cocyclic copurity.