  In this paper we show a method for computing the set of twists of a
non-singular projective curve defined over an arbitrary (perfect) field $k$.
The method is based on a correspondence between twists and solutions to a
Galois embedding problem. When in addition, this curve is non-hyperelliptic we
show how to compute equations for the twists. If $k = \mathbb{F}_{q}$ the
method then becomes an algorithm, since in this case, the Galois embedding
problems that appear are known how to be solved. As an example we compute the
set of twists of the non-hyperelliptic genus $6$ curve $x^{7}-y^{3}z^{4}-z^{7}
= 0$ when we consider it defined over a number field such that
$[k(\zeta_{21}):k] = 12$. For each twist equations are exhibited.
