A conformal automorphism \(\tau \) , of order \(n \geqslant 2\) , of a closed Riemann surface \(\mathcal {X}\) , of genus \(g \geqslant 2\) , which is central in \(\textrm{Aut}(\mathcal {X})\) and such that \(\mathcal {X}/\langle \tau \rangle \) has genus zero, is called a superelliptic automorphism of level n. If \(n=2\) , then \(\tau \) is the hyperelliptic involution and it is known to be unique. In this paper, for the case \(n \geqslant 3\) , we investigate the uniqueness of the cyclic group \(\langle \tau \rangle \) . Let \(\tau _{1}\) and \(\tau _{2}\) be two superelliptic automorphisms of level n of \(\mathcal {X}\) . If \(n \geqslant 3\) is odd, then \(\langle \tau _{1} \rangle =\langle \tau _{2} \rangle \) . If \(n \geqslant 2\) is even, the same uniqueness result holds, up to some explicit exceptional cases. We also provide conditions for these surfaces to be definable over their field of moduli.
