Multiferroic magnetoelectric materials, which simultaneously exhibit ferroelectricity and ferromagnetism, have recently stimulated a sharply increasing number of research activities for their scientific interest and significant technological promise in the novel multifunctional devices. Natural multiferroic single-phase compounds are rare, and their magnetoelectric responses are either relatively weak or occurs at temperatures too low for practical applications. In contrast, multiferroic composites, which incorporate both ferroelectric and ferri-/ferromagnetic phases, typically yield giant magnetoelectric coupling response above room temperature, which makes them ready for technological applications. This review of mostly recent activities begins with a brief summary of the historical perspective of the multiferroic magnetoelectric composites since its appearance in 1972. In such composites the magnetoelectric effect is generated as a product property of a magnetostrictive and a piezoelectric substance. An electric polarization is induced by a weak ac magnetic field oscillating in the presence of a dc bias field, and/or a magnetization polarization appears upon applying an electric field. So far, three kinds of bulk magnetoelectric composites have been investigated in experimental and theoretical, i.e., composites of (a) ferrite and piezoelectric ceramics (e.g., lead zirconate titanate), (b) magnetic metals/alloys (e.g., Terfenol-D and Metglas) and piezoelectric ceramics, and (c) Terfenol-D and piezoelectric ceramics and polymer. The elastic coupling interaction between the magnetostrictive phase and piezoelectric phase leads to giant magnetoelectric response of these magnetoelectric composites. For example, a Metglas/lead zirconate titanate fiber laminate has been found to exhibit the highest magnetoelectric coefficient, and in the vicinity of resonance, its magnetoelectric voltage coefficient as high as 102 V/cm Oe orders has been achieved, which exceeds the magnetoelectric response of single-phase compounds by many orders of magnitude. Of interest, motivated by on-chip integration in microelectronic devices, nanostructured composites of ferroelectric and magnetic oxides have recently been deposited in a film-on substrate geometry. The coupling interaction between nanosized ferroelectric and magnetic oxides is also responsible for the magnetoelectric effect in the nanostructures as was the case in those bulk composites. The availability of high-quality nanostructured composites makes it easier to tailor their properties through epitaxial strain, atomic-level engineering of chemistry, and interfacial coupling. In this review, we discuss these bulk and nanostructured magnetoelectric composites both in experimental and theoretical. From application viewpoint, microwave devices, sensors, transducers, and heterogeneous read/write devices are among the suggested technical implementations of the magnetoelectric composites. The review concludes with an outlook on the exciting future possibilities and scientific challenges in the field of multiferroic magnetoelectric composites.