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Highly sensitive AC magnetic field sensors are presented using magnetoelectric composites consisting of magnetostrictive and piezoelectric phases. They are offering passive nature, high sensitivity, large effect enhancement at mechanical resonance, and large linear dynamic range. Thin-film magnetoelectric 2-2 composites benefit from perfect coupling between the piezoelectric and magnetostrictive phases and from the reduction in size which is essential for high spatial resolution. Their design uses AlN and a plate capacitor or PZT with interdigital electrodes and magnetostrictive amorphous FeCoSiB single layers or exchanged biased multilayers. At mechanical resonance and depending on the geometry, extremely high ME coefficients of up to 9.7 kV/cm Oe in air and up to 19 kV/cm Oe under vacuum were obtained. To avoid external DC magnetic bias fields, composites consisting of exchanged biased multilayers serving as the magnetostrictive component with a maximum magnetoelectric coefficient at zero magnetic bias field are employed. Furthermore, the anisotropic response of these exchanged biased composites can be utilized for three-dimensional vector field sensing. Sensitivity and noise of the sensors revealed limits of detection as good as to 2.3 pT/Hz1/2 at mechanical resonance. Sensitivity between 0.1 and 1000 Hz outside resonance can be enhanced through frequency conversion using AC magnetic bias fields.