This thesis generalizes the moduli space of anti-self dual connections on principal bundles over nonorientable manifolds. Focusing on principal Pin4 bundles over a nonorientable base manifold X, we adapt the Hodge star operator for two forms by twisting it with an involution on the Lie algebra, resulting in an involution on the bundle of adjoint valued two forms. A connection is defined as *-ASD if its curvature resides in the (-1)-eigenspace of this generalized star operator. We analyze the moduli space M of *-ASD gauge equivalence classes of connections on P, beginning with a classification theorem for Pin4 bundles over a four-complex. We explore the local behavior of M, calculate its formal dimension, and examine singularities from reducible connections. A natural compactification of M is defined, and we compare M and its compactifications to the ASD moduli space on an SU2 bundle over the orientation cover of X. The thesis concludes with a discussion of the charge one moduli space on real projective space. Additionally, we prove theorems regarding the moduli space over an oriented four-manifold X with a smooth orientation-preserving action of a finite group, establishing criteria for the existence of non-empty fixed sets in the moduli space and conditions to avoid reducible ASD connections through slight equivariant perturbations of the Riemannian metric on X.
