A Cayley digraph $Γ$ over a finite group $G$ is said to be CI if for every Cayley digraph $Γ^\prime$ over $G$ isomorphic to $Γ$, there is an isomorphism from $Γ$ to $Γ^\prime$ which is at the same time an automorphism of $G$. In the present paper, we study a CI-property of normal Cayley digraphs over abelian groups, i.e. such Cayley digraphs $Γ$ that the group $G_r$ of all right translations of $G$ is normal in $Aut(Γ)$. At first, we reduce the case of an arbitrary abelian group to the case of an abelian $p$-group. Further, we obtain several results on CI-property of normal Cayley digraphs over abelian $p$-groups. In particular, we prove that every normal Cayley digraph over an abelian $p$-group of order at most $p^5$, where $p$ is an odd prime, is CI.