Recent developments in electronic structure theory have led to a new understanding of magnetic materials at the microscopic level. This enables a truly first-principles approach to investigations of technologically important magnetic materials. Among the advances treated here have been practical schemes for handling non-collinear magnetic systems, including relativity, and an understanding of the origins and role of orbital magnetism within band structure formalisms. This book provides deep theoretical insight into magnetism, mahneatic materials, and magnetic systems. It covers these recent developments with review articles by some of the main originators of these developments.