A topological group has the Rokhlin property if it contains an element whose conjugacy class is dense; an orientable manifold has the Rokhlin property if its group of orientation-preserving self-homeomorphisms has the Rokhlin property. Glasner and Weiss showed that every even-dimensional sphere has the Rokhlin property. We will give a characterization of all 2-manifolds with the Rokhlin property and present this result in context with recent results on homeomorphism groups/mapping class groups of self-similar 2-manifolds. Time permitting, we will discuss forthcoming work regarding 2-manifolds with the virtual Rokhlin property. This is joint work with Justin Lanier.