The key physical property of multiferroic materials is the existence of a coupling between magnetism and polarization, i.e. magnetoelectricity. The origin and manifestations of magnetoelectricity can be very different in the available plethora of multiferroic systems, with multiple possible mechanisms hidden behind the phenomena. In this Review, we describe the fundamental physics that causes magnetoelectricity from a theoretical viewpoint.The present review will focus on the main stream physical mechanisms in both single phase multiferroics and magnetoelectric heterostructures. The most recent tendencies addressing possible new magnetoelectric mechanisms will also be briefly outlined.Magnetism and electricity are two fundamental physical phenomena which have been widely covered in elementary textbooks of electromagnetism and have led to a broad technological revolution within human civilization. Even today, these two crucial subjects remain at the frontier of active research, and are still attracting considerable attention within the scientific community for their indispensable scientific value and possible applications. In solids, magnetism and electricity originate from the spin and the charge degrees of freedom, respectively. The crossover between these two fascinating topics has much grown in recent years and it has developed into an emergent branch of Condensed Matter Physics calledGenerally speaking, magnetoelectric effects can exist in many systems, even in some that are nonmagnetic. In fact, the first example of a magnetoelectric effect was observed by Röntgen in 1888 in a dielectric material, which was magnetized when moving through an electric field [10]. Much more recently, the surface state of topological insulators was predicted to manifest magnetoelectric effects [11]. However, to develop magnetoelectricity of a large magnitude, and as a consequence of more considerable practical value, multiferroics seem to be the best playground. In multiferroics, both magnetic moments and electric dipole moments can be ordered, inducing robust macroscopic quantities such as magnetization and polarization. Moreover, crucially for applications, both moments are coupled. Then, these macroscopic quantities may be mutually controlled, for example modifying the magnetization by an electric voltage or modifying the polarization by a magnetic field, which is particularly useful to design new devices, such as for storage and sensors.However, conceptually the mere existence of multiferroics is highly non trivial [12]. For most magnetic materials, the magnetic moments arise from unpaired electrons in partially occupied d orbitals and/or f orbitals. However, the spontaneous formation of a charge dipole usually needs empty d orbitals as a condition of having a coordinate bond, i.e. the so-called d 0 rule. Thus, the key ions involved in typical magnetic materials and those in polar materials are different, making these two areas of research nearly isolated from each other. However, in 2003 the discovery of a large polarizatio...