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In the past few decades, extensive research has been conducted on the magnetoelectric (ME) effect in single phase and composite materials. Dielectric polarization of a material under a magnetic field or an induced magnetization under an electric field requires the simultaneous presence of long-range ordering of magnetic moments and electric dipoles. Single phase materials suffer from the drawback that the ME effect is considerably weak even at low temperatures, limiting their applicability in practical devices. Better alternatives are ME composites that have large magnitudes of the ME voltage coefficient. The composites exploit the product property of the materials. The ME effect can be realized using composites consisting of individual piezomagnetic and piezoelectric phases or individual magnetostrictive and piezoelectric phases. In the past few years, our group has done extensive research on ME materials for magnetic field sensing applications and current measurement probes for high-power electric transmission systems. In this review article, we mainly emphasize our investigations of ME particulate composites and laminate composites and summarize the important results. The data reported in the literature are also compared for clarity. Based on these results, we establish the fact that magnetoelectric laminate composites (MLCs) made from the giant magnetostrictive material, Terfenol-D, and relaxor-based piezocrystals are far superior to the other contenders. The large ME voltage coefficient in MLCs was obtained because of the high piezoelectric voltage coefficient of the piezocrystals and large elastic compliances. In addition, an optimized thickness ratio between the piezoelectric and magnetostrictive phases and the direction of the magnetostriction also influence the magnitude of the ME coefficient.