Magnetoelectric (ME) effects in a ferromagnetic and piezoelectric composite are the changes in the polarization caused by a magnetic field or the changes in the magnetization caused by an electric field. These effects are aided by the mechanical deformation in the ferroic phases caused by the combination of magnetostriction and piezoelectricity. Interest in ME effects is due to a variety of physical phenomena they exhibit, as well as their potential applications in the creation of highly sensitive magnetic field sensors and other electronic devices. Linear ME effects in structures with layers of different ferroic materials have been studied extensively. However, nonlinear ME effects, which are caused by the nonlinearity of the magnetic, dielectric, and acoustic properties of ferromagnets and piezoelectrics, are less well understood. The purpose of this review is to summarize the current state of knowledge on nonlinear ME (NLME) effects in composite heterostructures and to discuss their potential applications. The review begins by discussing the characteristics of materials that are conductive to the occurrence of NLME effects and ferromagnetic-piezoelectric materials that are most commonly used to study such effects. The review then provides details on theoretical approaches to the description of NLME effects in heterostructures and experimental methods for studying these effects. Finally, the review presents a chronological overview of the experimentally observed NLME effects in composite structures excited by low-frequency and pulsed magnetic or electric fields. The review concludes with a discussion on the potential applications of NLME effects for highly sensitive magnetic field sensors.