  Let $S$ be a closed Riemann surface of genus $g\geq 2$ and let ${\rm Aut}(S)$
be its group of conformal automorphisms. It is well known that if either: (i)
${\rm Aut}(S)$ is trivial or (ii) $S/{\rm Aut}(S)$ is an orbifold of genus zero
with exactly three cone points, then $S$ is definable over its field of moduli
${\mathcal M}(S)$. In the complementary situation, explicit examples for which
${\mathcal M}(S)$ is not a field of definition are known. We provide upper
bounds for the minimal degree extension of ${\mathcal M}(S)$ by a field of
definition in terms of the quotient orbifold $S/{\rm Aut}(S)$.
