For spaces with an action of a group \(G\), there is a cohomology theory graded on real \(G\)-representations called Bredon cohomology which serves as an equivariant analogue of singular cohomology. This talk will explore a recent family of computations in \(RO(C_3)\)-graded Bredon cohomology where \(C_3\) is the cyclic group of order 3. We give an overview of a classification result for \(C_3\)-surfaces as well as demonstrate through small examples some of the techniques used to compute their cohomology.