These notes have no pretense of originality. The degree of mathematical rigour is low; arguments are given rather than mathematical proofs, but mathematical references are given where complete proofs can be found. Local coordinates will be used throughout in such a way that gluing the pieces together is more or less obvious. There is a serious motivation for introducing some of the mathematics that has been successfully used recently, however unusual: the challenge still holds to know whether there are non self-dual solutions of the full Yang-Mills equations, and eventually to find them all. Other non-linear conformal invariant field equations pose the same problem.