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Let $X$ be a smooth projective algebraic curve of genus $g\geq 2$ definedover a field $K$. We show that $X$ can be defined over its field of moduli ifit has odd signature, i.e. if the signature of the covering $X\to X/\Aut(X)$ isof type $(0;c_1,...,c_k)$, where some $c_i$ appears an odd number of times.This result is applied to $q$-gonal curves and to plane quartics. For $q$-gonalcurves, we prove that non-normal $q$-gonal curves can be defined over theirfield of moduli and we construct examples of normal $q$-gonal curves with fieldof moduli $\mathbb{R}$ that can not be defined over $\mathbb{R}$. For planequartics, we prove that they can be defined over their field of moduli if theautomorphism group is not isomorphic to either $C_2$ or $C_2\times C_2$.

Fields of moduli and fields of definition of odd signature curves