Two special classes of solutions to the Yang-Mills equations are studied in this thesis; Hermitian-Einstein connections on holomorphic bundles over Kahler manifolds, and self-dual connections on bundles over Riemannian 4-manifolds.

We give a new proof of a theorem of Narasimhan and Seshadri, which characterizes those holomorphic bundles over an algebraic curve admitting projectively flat connections, and describe a conjecture of Hitchin and Kobayashi that would extend this to Hermitian-Einstein connections over any smooth projective variety. This conjecture is proved to be true for the simplest interesting case: bundles of rank 2 over ℙ².

Moduli spaces of self-dual connections are studied from the point of view of differential topology, For bundles of Chern class -1 over a simply connected 4-manifold this moduli space can be compactified in a straightforward way and is, in a generic sense, an orientable manifold with quotient singularities. Applying the theory of cobordism to this moduli space we deduce that there are severe constraints on the matrices which can be realised by the intersection pairing on the second homology group of a smooth 4-manifold.