In this book, the author's goal is to provide an introduction to some of the analytic underpinnings for the geometry of antiself duality in 4dimensions. Antiself duality is rather special to 4dimensions and the imposition of this condition on curvatures of connections on vector bundles and on curvatures of Riemannian metrics has resulted in some spectacular mathematics. The book reviews some basic geometry, but it is assumed that the reader has a general background in differential geometry (as would be obtained by reading a standard text on the subject). Some of the fundamental references include Atiyah, Hitchin and Singer, Freed and Uhlenbeck, Donaldson and Kronheimer, and Kronheimer and Mrowka. The last chapter contains open problems and conjectures. Readership Graduate students and research mathematicians interested in differential geometry. Reviews "Provides an excellent introduction to the application of certain analytic techniques to problems in differential geometry ... the casual style in which this book is written together with the straightforward explanations of the key ideas underlying the theory makes it an excellent source for those wishing to learn about these basic techniques ... a perfect balance seems to have been struck in the choice between what to include and what to refer the reader elsewhere for."  Mathematical Reviews "Easy to read ... well written in a pleasant, informal style, with occasional humour ... should be accessible to graduate students in differential geometry and others."  Bulletin of the LMS Table of Contents  Introduction
 The antiself dual equations
 Grafting theorems
 Deformations to antiself duality I
 Deformations to antiself duality II
 Metrics with \(W_+\equiv 0\)
 Grafting metrics
 Deforming the metric
 Strategy for connect sums
 Open questions
 References
