  As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$
can be described by a non-singular complex projective algebraic curve $C$. A
field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may
choose $C$ to be defined by polynomials in $D[x_0, \ldots, x_n]$. The field of
moduli of $S$ is ${\mathbb R}$ if and only if $S$ admits an anticonformal
automorphism. In the case that the field of moduli of $S$ is ${\mathbb R}$,
then $S$ can be defined over the field of moduli if and only if $S$ admits an
anticonformal involution. It may happen that the field of moduli is not a field
of definition.
  In this paper, we consider certain class of closed Riemann surfaces, called
generalized Fermat curves. These surfaces are the highest Abelian branched
cover of certain orbifolds. In this class of Riemann surfaces, we study the
problem of deciding when the field of moduli is ${\mathbb R}$ and when, in such
a case, it is a field of definition.
