Abstract: To any suitable geometric space (closed smooth manifold), one can associate a real number called the Yamabe invariant which arises from considerations in Riemannian geometry. For surfaces, this number is a familiar quantity, but in higher dimensions, it is less understood. However, as we will see, more can be said if one restricts to those spaces which admit a complex structure, e.g., orientable surfaces. This talk is partly based on joint work with Claude LeBrun.