The dynamics of current-driven magnetic domain walls and skyrmions are both topics of high interest in the magnetism community due to the possibility of employing them in novel high-density non-volatile memory devices. One such device is the well-known racetrack memory, where the information is stored in a ferromagnetic nanowire as compared to the conventional harddisk memory, while the operation is mediated via the injection of current to the nanowire. In a domain wall-based racetrack memory, the information is represented by magnatic domains that are separated by domain walls, while in a skyrmionbased racetrack memory, the information is represented by the presence of the skyrmions themselves. However, to be able to realize such devices, it is very important to have a complete understanding of both the domain wall and dynamics under the application of current. At present, the investigation of domain wall dynamics is mainly focused on finding additional driving mechanisms to improve its mobility. In the case of skyrmion dynamics, the research is still in the very early stage, although it has been shown that it is possible to inject individual skyrmions individually which brings skyrmion-based devices closer to realization. In this thesis, the dynamics of current driven magnetic domain walls and skyrmions in various nanostructures are investigated. We show by micromagnetic simulations that it is possible to drive multiple domain walls in a multi-nanowire system by just applying current to one of the nanowires. The phenomenon is made possible due to the magnetostatic coupling between the domain walls. When materials with in-plane anisotropy are considered, the coupling is realized when the nanowires are placed parallel to each other, and the technique can be further utilized to drive several domain walls in the current-free nanowire. When materials with strong perpendicular magnetization anisotropy are used, the phenomenon is realized by stacking the nanowires vertically, and the coupling between the