This short paper presents characterisations of normal arc-transitive circulants and arc-transitive normal circulants, that is, for a connected arc-transitive circulant $Γ=\Cay(C,S)$, it is shown that 1. Aut(C,S) is transitive on S if and only if each element of S has order n; 2. $AutΓ\rhd C$ if and only if S does not contain a coset of any subgroup. This completes the classification of arc-transitive circulants given by Li-Xia-Zhou.