  Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface
and $G$ be a finite subgroup in $\operatorname{Aut}(X)$. In this paper we study
when the quotient surface $X / G$ can be non-rational over $\Bbbk$. Obviously,
if there are no smooth $\Bbbk$-points on $X / G$ then it is not
$\Bbbk$-rational. Therefore under assumption that the set of smooth
$\Bbbk$-points on $X / G$ is not empty we show that there are few possibilities
for non-$\Bbbk$-rational quotients.
  The quotients of del Pezzo surfaces of degree $2$ and greater are considered
in the author's previous papers. In this paper we study the quotients of del
Pezzo surfaces of degree $1$. We show that they can be non-$\Bbbk$-rational
only for the trivial group or cyclic groups of order $2$, $3$ and $6$. For the
trivial group and the group of order $2$ we show that both $X$ and $X / G$ are
not $\Bbbk$-rational if the $G$-invariant Picard number of $X$ is $1$. For the
groups of order $3$ and $6$ we construct examples of both $\Bbbk$-rational and
non-$\Bbbk$-rational quotients of both $\Bbbk$-rational and
non-$\Bbbk$-rational del Pezzo surfaces of degree $1$ such that the
$G$-invariant Picard number of $X$ is $1$.
  As a result of complete classification of non-$\Bbbk$-rational quotients of
del Pezzo surfaces we classify surfaces that are birationally equivalent to
quotients of $\Bbbk$-rational surfaces, and obtain some corollaries concerning
fields of invariants of $\Bbbk(x , y)$.
