  A surface with an involution can be viewed as a $C_2$-space where $C_2$ is
the cyclic group of order two. Using the classification of $C_2$-surfaces given
by Dugger, we compute the $RO(C_2)$-graded Bredon cohomology of all
$C_2$-surfaces in constant $\mathbb{Z}/2$ coefficients as modules over the
cohomology of a point. We show the cohomology depends only on three numerical
invariants in the nonfree case, and only on two numerical invariants in the
free case.
