The main drivers of space weather, solar flares and coronal mass ejections, are thought to originate from active regions on the surface of the Sun. The mechanisms by which active regions produce these eruptive events remains unclear. In this thesis, numerous mathematical methods are developed to characterise the complexity of active region magnetic fields on the Sun. In the first section of the thesis, a box-counting method was used to measure the temporal evolution of the multifractal parameters of a sample of active regions. These results are compared to other multi-scale methods and changes in the temporal evolution of the multifractal spectrum are found to be associated with a regions flaring potential. As an expansion of the box-counting method, a wavelet-based multifractal method, the wavelet transform modulus methods (WTMM), was similarly used to characterise the changing properties of active region magnetic fields. A study of the multifractal properties of active and quiet Sun magnetic fields showed them to be statistically distinct and separable in wavelet transform space. As such, a segmentation procedure was developed to accurately recover the multifractal parameters of active region magnetic fields from the surrounding quiet Sun. Additionally, the temporal evolution of the fractal dimension and Holder exponent of active regions were examined for a possible relation to flaring. The complexity of the coronal field was investigated using extrapolation techniques. A linear force-free extrapolation method was developed to examine the changing topology of active region magnetic fields. Comparisons were made to EUV observations of the solar corona so as to constrain the extrapolations. Using the twin perspectives of the Solar TErrestrial RElations Observatory (STEREO) spacecraft the 3D topology of a coronal loop was identified and modelled with time. Changes in the amount of twist, free energy and connectivity of the loop were related to magnetic flux emergence in NOAA 10956. TARA (Trinity’s Access to Research Archive) has a robust takedown policy. Please contact us if you have any concerns: rssadmin@tcd.ie