Perfect state transfer (PST) has great significance due to its applications in quantum information processing and quantum computation. In this paper we present a characterization of the connected simple Cayley graph Γ = Cay ( G , S ) having PST, where G is an abelian group and S is a non-empty set of G . We show that many previous results on periodicity and existence of PST of circulant graphs (where the underlying group G is cyclic) and cubelike graphs ( G = ( F 2 n , + ) ) can be derived or generalized to arbitrary abelian case in unified and more simple ways from our characterization. We also get several new results including answers to some questions raised before.