This thesis investigates the topology of the magnetic field in the solar corona. It is important have an understanding of how the highly complex coronal magnetic field behaves in order to study many fundamental coronal phenomena, such as coronal heating events, solar flares and polar plumes. The magnetic fields due to three or four discrete sources are investigated and the corresponding topological states are found. The locations of these states in parameter space is calculated and the bifurcations between states are analysed. A complete analysis has been undertaken for the three-source case and a selective one for the four-source case in order to identify new non-generic behaviour. The thesis goes on to study the topological behaviour of a coronal bright point. Different phases during the lifetime of the bright point are identified and the responsible topological behaviour due to the movement of the magnetic fragments in the photosphere is discussed.