An important class of metrics in dimension four known as anti-self-dual and self-dual metrics is studied. In particular, the focus of this thesis is on the moduli space of anti-self-dual and self-dual conformal classes on orbifolds and questions relating to the existence of such metrics. In Part I an index formula for the anti-self-dual deformation complex of anti-self-dual orbifolds with a cyclic quotient singularity is proved. Then we present two applications of this theorem; one to study deformations of a family of scalar-flat Kahler toric asymptotically locally Euclidean spaces and another to study deformations of the canonical Bochner-Kahler metric on any weighted projective space. In Part II index formulas for the anti-self-dual and self-dual deformation complexes of compact 4-manifolds with orbifold-cone metrics are proved. These metrics are a subclass of edge-cone metrics, which have recently been of much interest in Kahler geometry. We then give a variety applications of these formulas to study the moduli space of anti-self-dual and self-dual conformal classes of several compact 4-manifolds with orbifold-cone metrics, and address questions relating to the existence of these metrics.
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