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Nano- and micro-electromechanical systems (N/MEMSs) are traditionally based on electrostatic or piezoelectric coupling, which couples electrical and mechanical energy through acoustic resonator structures. Most recently, N/MEMS devices based on magnetoelectrics are gaining much attention. Unlike electrostatic or piezoelectric N/MEMS that rely on an AC electric field or voltage excitation, magnetoelecric N/MEMS rely on the electromechanical resonance of a magnetostrictive/piezoelectric bilayer heterostructure exhibiting a strong strain-mediated magnetoelectric coupling under the excitation of a magnetic field and/or electric field. As a consequence, magnetoelectric N/MEMS enable unprecedented new applications, ranging from magnetoelectric sensors, ultra-compact magnetoelectric antennas, etc. This Tutorial will first outline the fundamental principles of piezoelectric materials, resonator design, specifically different acoustic modes, and piezoelectric-based N/MEMS applications, i.e., radio frequency front end filters and infrared radiation sensors. We will then provide an overview of magnetoelectric materials and N/MEMS focusing on the governing physics of the magnetoelectric effect, magnetic material properties for achieving high magnetoelectric coupling, state-of-the-art magnetoelectric N/MEMS devices, and their respective applications.