Quotients of rational surfaces Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$.
Assuming that $X$ is rational over $k$, is the quotient $X/G$ always rational?
If $k=\mathbb{C}$, we can use Castelnuovo's theorem and see that $X/G$ is unirational and hence rational. If $k=\mathbb{R}$, then $X/G$ is geometrically rational and also connected for the transcendental topology, and is thus rational.
But what happens for a general $k$, in particular when $k=\mathbb{Q}$?
 A: This is not always true, and cubic threefolds give a counterexample over the field $k=\mathbb{C}(t)$.  Let $\mathcal{Y}$ be a smooth cubic hypersurface in $\mathbb{P}^4_{\mathbb{C}}$.  Let $L\subset \mathcal{Y}$ be a line.  Denote by $\mathcal{X}$ the (locally closed) subvariety of $\mathcal{Y}\times L$ parameterizing pairs $(y,p)$ such that the intersection of $\text{Span}(L,y)$ with $\mathcal{Y}$ is a plane cubic $L\cup C$, where $C$ is a plane conic intersecting $L$ transversally at $p$.  This condition on the conic $C$ is valid for all $y$ in a dense open subset of $\mathcal{Y}\setminus L$.  Define an involution, $$ i:\mathcal{X} \to \mathcal{X}, \ i(y,p) = (y,q), $$ where $C\cap L$ equals $\{ p,q \}$.   This involution defines an action on $\mathcal{Y}$ by the cyclic group $G$ of order.  The quotient is the (dense, open) image $U$ of the projection $\text{pr}_{\mathcal{Y}}:\mathcal{X}\to \mathcal{Y}$.
How does this give a counterexample for surfaces?  Let $\Pi$ be a linear $2$-plane containing $L$.  Let $$\pi:(\mathbb{P}^3_{\mathbb{C}} \setminus \Pi) \to \mathbb{P}^1_{\mathbb{C}}$$
be linear projection away from $\Pi$.  Let $U_\Pi$ be $U\setminus \Pi$, and let $\mathcal{X}_{\Pi}$ be the inverse image of $U_\Pi$ in $\mathcal{X}$.  Of course this is a $G$-invariant, dense, open subset of $\mathcal{X}$.  The claim is that a general fiber of $f\circ \text{pr}_{\mathcal{Y}}:\mathcal{X}_{\Pi} \to \mathbb{P}^1$ is a rational surface.  Then letting $k$ be the function field $\mathbb{P}^1_{\mathbb{C}}$, and letting $Y$ and $X$ be the generic fiber of $f$, resp. $f\circ \text{pr}_{\mathcal{X}}$, this gives a counterexample.
Consider the morphism $$(f\circ \text{pr}_{\mathcal{Y}}, \text{pr}_{L}): \mathcal{X}_{\Pi} \to \mathbb{P}^1_{\mathbb{C}} \times_{\mathbb{C}} L.$$  A general point of the target parameterizes a pair $([H],p)$, where $H$ is a hyperplane in $\mathbb{P}^3_{\mathbb{C}}$ containing $\Pi$, and where $p$ is a point of $L$.  Consider the "projective linear" tangent space to $X$ at $p$, i.e., the unique hyperplane $\Sigma$ in $\mathbb{P}^3_{\mathbb{C}}$ with maximal order of contact with $X$ at $p$.  Then $\Sigma$ contains $L$.  The intersection of $\Sigma$ and $H$ is a linear $2$-plane $\Xi$ that contains $L$.  If $p$ and $H$ are general then $\Xi$ is not equal to $\Pi$, and the intersection of $\Xi$ with $\mathcal{Y}$ is a plane cubic $L\cup C$, where $C$ is a plane conic that intersects $L$ transversally at $p$ and $i(p)$.  Thus the fiber of $(f\circ \text{pr}_{\mathcal{Y}}, \text{pr}_{L})$ over $([H],p)$ is $C\setminus \{p,i(p)\}$.  Therefore, at least after passing to a dense open subset of the target, the morphism $(f\circ \text{pr}_{\mathcal{Y}}, \text{pr}_{L})$ is a dense open subset of a conic bundle.  Moreover, this conic bundle has a section; namely send $([H],p)$ to the point $p$ of the conic $C$.  A conic bundle with a section is birational to $\mathbb{P}^1$ over the base.  Thus the composite morphism $$f\circ \text{pr}_{\mathcal{Y}}:\mathcal{X}_\Pi \to \mathbb{P}^1_{\mathbb{C}} $$ is birational to $$\text{pr}_1:\mathbb{P}^1_{\mathbb{C}} \times_{\mathbb{C}} \mathbb{P}^1_{\mathbb{C}}\times_{\mathbb{C}} L \to \mathbb{P}^1_{\mathbb{C}}.$$  
If memory serves, this description of $\mathcal{X}$ as a conic bundle is described in the appendix to Clemens and Griffiths where they explain Mumford's Prym construction.
Edit.  Of course the point is that the Clemens-Griffiths theorem proves that $\mathcal{Y}$ is not rational over $\mathbb{C}$.  If the generic fiber $Y$ of $f$ were rational over $k=\mathbb{C}(t)$, then $\mathcal{Y}$ would be rational over $\mathbb{C}$.
Edit.   I decided to add the following comment to the answer.  In his book "Cubic Forms", Manin seems to give examples of quartic del Pezzo surfaces $Y$ over number fields that have a rational point, that have a degree $2$ double-cover $X$ that is rational (so that $Y$ is $X/G$ for $G$ a cyclic group of order $2$), yet with $Y$ irrational.   The reference is Theorem IV.29.2, Theorm IV.29.4 and Remark IV.29.4.1, pp. 157-158 with r=5, and also Section IV.31, pp. 174--182. 
A: Just to round out the picture: if the characteristic of $k$ is positive and $G$ is a finite, but non-reduced group scheme (for example, the infinitesimal group scheme $\mu_p$ of $p$.the roots of unity), then the quotient $X/G$ need not even have Kodaira dimension $-\infty$ after desingularization. Moreover, if $G$ is not linearly reductive (for example, the infinitesmial group scheme $\alpha_p$), then a resolution of singularities $f:Y\to X/G$ need not satisfy $R^if_\ast{\mathcal O}_Y=0$ for $i\geq1$. Both phenomenons occur already if $\dim X=2$ - for example, there are many examples of ``unirational surfaces of general type'' in positive characteristic.
