Given a space with a \(C_2\)-action, we can consider the \(RO(C_2)\)-graded Bredon cohomology of the space. Dugger recently classified all \(C_2\)-surfaces up to equivariant isomorphism using equivariant surgery. This classification lends itself nicely to Bredon cohomology computations, and computations have now been done for all \(C_2\)-surfaces in coefficients given by both the constant \(\underline{\mathbb{Z}}\) and \(\underline{\mathbb{Z}/2}\) Mackey functors. In the case of \(\underline{\mathbb{Z}/2}\)-coefficients, the answer is nice and depends only on the fixed set and the singular cohomology of the underlying space. In this talk, we will introduce this cohomology theory and show how equivariant surgery can be used in computations for \(C_2\)-surfaces. We will then explore the answers in both of the coefficient systems mentioned above.