This an introductory book on Self-Dual Riemannian 4- Manifolds. Self-Dual metrics are special type of metrics which provide solution to the "Optimal Metric" problem. Under a vanishing hypothesis, Donaldson and Friedman proved that the connected sum of two self-dual Riemannian 4-Manifolds is again self-dual. Here we prove that the same result can be extended over to the positive scalar curvature case. The idea is to use Leray spectral sequence. Secondly we give an example of a 4-manifold with b+ = 0 admitting a scalar-flat anti-self-dual metric. Finally we present an application of the Geometric Invariant Theory(GIT) for Toric Varieties to the Einstein-Weyl Geometry.

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