We theoretically study the equilibrium and dynamic properties of nanoscale magnetic tunnel junctions (MTJs) and magnetic wires, in which an electric field controls the magnetic anisotropy through spin-orbit coupling. By performing micromagnetic simulations, we construct a rich phase diagram and find that, in particular, the equilibrium magnetic textures can be tuned between Neel and Bloch domain walls in an elliptical MTJ. Furthermore, we develop a phenomenological model of a quasi-one-dimensional domain wall confined by a parabolic potential and show that, near the Neel-to-Bloch-wall transition, a pulsed electric field induces precessional domain-wall motion which can be used to reverse the chirality of a Neel wall and even depin it. This domain-wall motion controlled by electric fields, in lieu of applied current, may provide a model for ultra-low-power domain-wall memory and logic devices.