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We theoretically study equilibrium and dynamic properties of nanosized magnetic skyrmions in thin magnetic films with broken inversion symmetry, where an electric field couples to magnetization via spin-orbit coupling. Based on a symmetry-based phenomenology and micromagnetic simulations we show that this electric-field coupling, via renormalizing the micromagnetic energy, modifies the equilibrium properties of the skyrmion. This change, in turn, results in a significant alteration of the current-induced skyrmion motion. Particularly, the speed and direction of the skyrmion can be manipulated by designing a desired energy landscape electrically, which we describe within Thiele's analytical model and demonstrate in micromagnetic simulations including electric-field-controlled magnetic anisotropy. We additionally use this electric-field control to construct gates for controlling skyrmion motion exhibiting a transistorlike and multiplexerlike function. The proposed electric-field effect can thus provide a low-energy electrical knob to extend the reach of information processing with skyrmions. [ABSTRACT FROM AUTHOR]