Over recent years the investigation into the magnetic behaviour of nanostructured p ermalloy has become more advanced due to improvements in numerical micromag-netic methods on the theoretical side and high accuracy electron-beam lithography methods exp erimentally. The interest in such structures of magnetic material is increasing mainly due to the possible potential use in future high-density magnetic storage media a pplications. When the material is discretised into a nanoelement structure at the sub micron level t heoretical micromagnetic techniques may be employed in order to investi-gate the magnetization behaviour. This thesis describes a theoretical study of the hysteresis and domain behav iour in thin film permalloy nanoelements. To carry out our investigations we have developed a dynamical micromagnetic model based on t he use of the finite element method. The results presented in this t hesis begin with a test of the performance of our model. We then proceed with an investigation into the effect of size, elongation and geometry on the transition states for single nanoelements. The investigation is then extended to look at t he magnetization behaviour of arrays of interacting nanoelements in relation to their separation and material propert ies . The reversal mechanism of the arrays is very sensitive to the degree of disorder. In the case of an aligned uniaxial anisotropy a highly symmetric cooperative switch-ing mechanism is observed. A large anisotropy has t he effect of stabilizing states during the reversal process leading to distinctive switching. A random anisotropy breaks this high symmetry sufficiently to reduce the cooperative switching leading to a relatively random reversal of individual elements. The theoretical predictions are compared with experimental observations.