The (cyclic) Nielsen realization problem for a closed, oriented manifold asks whether any mapping class of finite order m can be represented by a homeomorphism of order m. In this talk I will discuss two results about the Nielsen realization problem for del Pezzo surfaces M^4. First, I will explain a classification of order-2 elements of the topological mapping class group Mod(M^4) and deduce the realizability of order-2 mapping classes by diffeomorphisms. I will also discuss joint work in progress with Tudur Lewis and Sidhanth Raman comparing the smooth, complex, and metric versions of the Nielsen realization problem for certain ("irreducible'') finite-order elements of Mod(M^4).