  A superelliptic curve $\X$ of genus $g\geq 2$ is not necessarily defined over
its field of moduli but it can be defined over a quadratic extension of it.
While a lot of work has been done by many authors to determine which
hyperelliptic curves are defined over their field of moduli, less is known for
superelliptic curves. In this paper we observe that if the reduced group of a
genus $g\geq 2$ superelliptic curve $\X$ is different from the trivial or
cyclic group, then $\X$ can be defined over its field of moduli; in the cyclic
situation we provide a sufficient condition for this to happen. We also
determine those families of superelliptic curves of genus at most $10$ which
might not be definable over their field of moduli.
