The choice of a metric (up to conformal reascaling) on an orientable, four dimensional manifold defines a splitting of the 2nd exterior power of the cotangent bundle into a direct sum of two oriented, 3-plane bundles. There is a corresponding, canonical splitting of the Weyl curvature tensor into a direct sum of two pieces. With the preceding understood, an orientable, four dimensional, Riemannian manifold will be said to be an anti-self dual geometry when its metric is such that one of these two pieces of the Weyl curvature tensor vanihses. This lecture begins by describing some of the recent existence theorems for anti-self dual geometries. The lecture continues with a description of the correspondence (due to Penrose) between anti-self dual geometries and complex 3-folds. With the preceding serving as a first movement, the lecture concludes with a presentation of some amusing and possibly important open problems and conjectures concerning anti-self dual geometries and related phenomena in three and four dimensional geometry.