In the last few years, revolutionary advances in experimental techniques and spectacular increases in computer power have offered unique opportunities to develop a much more profound understanding of atomic collisions processes. Part of the excitement of the area lies in the fact that modern measurements, because of their range and sophistication, constantly present new challenges to theory. The theoretical response to these new experiments has been to produce a plethora of new approximations and numerical techniques. The current book gives a relatively simple and coherent modern account of the theory of electron and positron scattering, where the essential Physics is clearly explained and where the theoretical methods used are firmly placed in the context of this Physics. Thus, the book could be read with profit by someone studying the theory for the first time, and it even carries the development far enough for the reader to be prepared to begin research in the field. The authors have taken pains to maintain a consistent philosophy of approximation, from the simplest potential scattering through to studies of the multiple ionization of complex targets.