In this thesis some solutions to classical Euclidean Yang-Mills theory are considered and presented with emphasis on self-dual SU(2) solutions. Chapter one is a brief introduction to the subject. In chapter two the possibility of solution via inversion of the dynamic equation to obtain in terms or Fuv and then imposing a self-consistency requirement is considered. Chapter three deals with the extension of Witten's method of solution through cylindrically symmetric ansatze to self- dual SU(3) theories. In Chapter four the invariances and Backlund-type transformations inherent in a self-dual SU(n) theory are investigated and these methods are used in Chapter five to present an analytic method for constructing all self-dual SU(2) solutions.