We review the details of domain wall (DW) propagation due to spin-polarized currents that could potentially be used in magnetic data storage devices based on domains and DWs. We discuss briefly the basics of the underlying spin torque effect and show how the two torques arising from the interaction between the spin-polarized charge carriers and the magnetization lead to complex dynamics of a spin texture such as a DW. By direct imaging we show how confined DWs in nanowires can be displaced using currents in in-plane soft-magnetic materials, and that when using short pulses, fast velocities can be attained. For high-anisotropy out-of-plane magnetized wires with narrow DWs we present approaches to deducing the torque terms and show that in these materials potentially more efficient domain wall motion could be achieved.