In this thesis, current driven domain-wall motion is studied using atomistic simulations with the exchange coupling modeled by the Heisenberg Hamiltonian under the nearest-neighbor approximation. The investigations may be divided into two parts, each concerned with how different aspects of the systems affect the domain-wall motion. The first part deals with domain-wall width dependence of the velocity in a three dimensional geometry with simple cubic crystal structure and uniaxial anisotropy. Results from this part showed that the velocity has a minor domain-wall width dependence. For a fixed current density, the velocity increased with domain-wall width, though only from 61.5 a/ns to 64.5 a/ns as the domain-wall width was increased from 3 to 25 atoms. The second part of the investigations deals with phenomena involving mixed cubic and uniaxial anisotropy, the non-adiabaticity parameter as well as the geometry of the system. The discussion includes an account of how the spin-transfer and cubic anisotropy torques contribute to the motion for different values of the non-adiabaticity parameter. In comparing a one dimensional atomic chain and a three dimensional system with simple cubic crystal structure, but otherwise with the same material properties, results showed a difference in how the two systems responded to currents. This difference is not accounted for by the micromagnetic theory, and its origin was unable to be determined.