  In this paper, we study the $G$-representation and character varieties of
non-orientable closed surfaces. By means of a geometric method based on a
Topological Quantum Field Theory (TQFT), we compute the virtual classes of
these varieties in the Grothendieck ring of varieties for $G$ equal to
$\textrm{AGL}_1$ and $\textrm{SL}_2$. This method was already known and used in
the case of orientable closed surfaces, and we extend it to the case of
non-orientable surfaces. Furthermore, we provide a practical approach for
explicitly computing the TQFT, allowing for more simplified and structured
computations. Finally, for $G = \textrm{SL}_2$ we describe and explain the
relationship between the representation varieties of the orientable and
non-orientable closed surfaces.
