  Given a perfect field $k$ with algebraic closure $\overline{k}$ and a variety
$X$ over $\overline{k}$, the field of moduli of $X$ is the subfield of
$\overline{k}$ of elements fixed by field automorphisms
$\gamma\in\operatorname{Gal}(\overline{k}/k)$ such that the twist $X_{\gamma}$
is isomorphic to $X$. The field of moduli is contained in all subextensions
$k\subset k'\subset\overline{k}$ such that $X$ descends to $k'$. In this paper
we extend the formalism, and define the field of moduli when $k$ is not
perfect.
  Furthermore, D\`ebes and Emsalem identified a condition that ensures that a
smooth curve is defined over its field of moduli, and prove that a smooth curve
with a marked point is always defined over its field of moduli. Our main
theorem is a generalization of these results that applies to higher dimensional
varieties, and to varieties with additional structures.
  In order to apply this, we study the problem of when a rational point of a
variety with quotient singularities lifts to a resolution. As a consequence, we
prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such
that $\operatorname{Aut}(X,p)$ is finite, \'etale and of degree prime to $d!$
is defined over its field of moduli.
