The chapter on “Advances in magnetoelectric materials and their application” updates and complements the review on “Magnetoelectricity” contained in Volume 19 of the present series. Interest in single-phase and composite magnetoelectric multiferroics continues to rise. Current review provides a portrait of the state of the art of magnetoelectric materials, both from the standpoint of basic science and view of applications. In the first part of this edition, the authors deliver a friendly explanation of magnetoelectric multiferroicity from the atomic to mesoscopic scale. Several geometrical aids for understanding the physics of magnetoelectricity are supplied. A representative sample of single-phase magnetoelectrics, considered important today, is characterized. Various types of crystal structures, frequently related to perovskites, occupy much of the attention in the literature and are described in the review. Besides obliged BiFeO3, double perovskites, Aurivillius (layered perovskites) phases, pseudo-cubic perovskites, orthoferrites, and other cases are analyzed. The effect of crystallographic texture on the magnetoelectric performance of polycrystalline single-phased materials is discussed. The wide range of responses that are generated by texture, and its link with magnetic symmetry, is presented. Magnetoelectric composites currently reaffirm themselves as the magnetoelectric materials with best figures of merit and with greater versatility when seeking the ideal material for an intended application. A current research tendency is to take advantage of (electromechanical, magnetic, and magnetoacoustic) resonances to enhance the magnetoelectric performance. The field of applications for magnetoelectricity is growing interestingly. In addition to the clearly identified fields of sensors development, magnetoelectric memories and energy harvesters, new trends in spintronics and medicine are worth remarking for their striking potential. Definitely, regarding the applications of magnetoelectricity, the best is yet to come. © 2015 Elsevier B.V.