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Giant magnetoelectric (ME) effects have been investigated recently in composites of magnetostrictive and piezoelectric materials. The usage of highly strictive piezoelectric and magnetostrictive materials as well as an increased understanding of their elastic coupling leads to magnetoelectric (ME) responses that are much higher than those published in the 1970s. Magnetic field sensors based on the ME effect promise sensitivities approaching those of low-temperature SQUID sensors. This paper concentrates on ME devices that were fabricated by sputtering magnetostrictive Fe-Co/Tb-Fe multilayers on piezoelectric PMN-PT single crystals. Intrinsic and induced magnetic anisotropies lead to a pronounced directional dependence of the ME effect. Thin magnetic films display anisotropies that are induced by the demagnetization field that can reduce the "out of plane" ME effects. "In-plane" ME effects can be enhanced by annealing and field-cooling leading to a pronounced easy and hard direction. In Fe-Co/Tb-Fe multilayers the magnetostriction along the easy axis vanishes and the ME effect is maximized along the hard axis. Therefore, a combination of three individual sensors realizes a magnetic vector field sensor.