In the quest for higher data density in information technology manipulation of magnetization by other means than magnetic fields has become an important challenge. This lead to a startling revival of the magnetoelectric effect, which characterizes induction of a polarization by a magnetic field or of a magnetization by an electric field. The magnetoelectric crosslink of material properties opens just those degrees of freedom which are needed for the mutual control of magnetic and electric states. The book gives a state-of-the-art review on magnetoelectrics research, classifies current research tendencies, and points out possible future trends. Novel compounds and growth techniques and new theoretical concepts for the understanding of magnetoelectric coupling phenomena are introduced. Highlights are the discovery of "gigantic" magnetoelectric effects which are strong enough to trigger electric or magnetic phase transitions; the concept of magnetochirality; and development "structural" magnetoelectric effects in artificial multiphase compounds. The book is addressed to condensed-matter physicists with a particular focus on experts in highly correlated systems.