A rational ruled surface is a P^1 bundle over P^1. The infinitesimal deformation theory of these surfaces is quite simple and has been well understood since Kodaira and Spencer first started developing the subject in the early 60s. I will review this theory and discuss a possible global theory as well as its connection with the topology of the group of symplectomorphisms of S^2 bundles over S^2. This is based on joint work with Miguel Abreu (IST) and Nitu Kitchloo (UC San Diego).