  We study skyrmions in magnetic thin films with structural inversion asymmetry
perpendicular to the film plane. We determine the magnetization texture of a
single skyrmion and its dependence on the strength of the Dzyaloshinskii-Moriya
interaction relative to the magnetostatic energy. Furthermore, we construct a
phenomenological model that describes the interaction between the motion of
skyrmions and electric currents to lowest order in spin-orbit coupling. We
estimate the experimental verifiable velocities for current-driven motion of
skyrmion textures based on available results obtained from domain walls
dynamics.
