We present our recent theoretical results on the magnetization dynamics in ferromagnetic and multiferroic heterostructures, where the magnetic anisotropy of a ferromagnetic nanolayer varies with the electric field created in adjacent dielectric or ferroelectric film. First, we describe the magnetization oscillations excited by microwave voltages and show that this dynamic magnetoelectric effect acquires anomalous characteristics near thickness-induced spin reorientation transitions. Second, we consider the influence of electric-field-dependent interfacial anisotropy on the current-driven magnetization reversal in magnetic tunnel junctions (MTJs). The theory explains easy magnetization switching recently discovered in CoFeB/MgO/CoFeB junctions and provides a recipe for the fabrication of MTJs suitable for memory applications. Finally, we describe the magnetization reorientations induced by the polarization switching in a ferroelectric film adjoining a ferromagnetic nanolayer.