The magnetic properties of nanometre-scale structures are of fundamental scientific interest and have the potential to play a major role in future data storage technologies. In particular, arrays of small magnetic elements, also called bit-patterned media, are one of the most promising candidates for the future generation of data storage devices. In this thesis we study potential bit patterned element geometries which are below 1 micrometre in size. Their magnetic behaviour is hard to predict using analytical methods and computer simulations are the principal tool for in-depth analysis. The relevant micromagnetic equations are solved using the combined Finite Element/Boundary Element method, and finite differences. Patterned media are (quasi) periodic arrangements of identical objects, with each object typically representing one bit. While one or some of these objects can be simulated with today’s simulation capabilities, the investigation of arrays with hundreds of objects requires novel simulation methods. To deal with such large arrays we introduce and evaluate the new “macro geometry” approach. In most real samples this is superior to using conventional periodic boundary conditions as it takes account of the macroscopic shape of the sample. The micromagnetic simulation package Nmag developed at Southampton has been extended to provide the macro geometry capabilities, and subsequently used to study demagnetising effects between the elements of triangular ring arrays. We find that in a square array of 50-nm size triangular elements these effects are governed by the first and second nearest neighbours and can be considered negligible when the spacing between the rings is larger than 30 nm. We also study the transport properties via the Anisotropic Magneto Resistance (AMR) signal of connected rings arrays using the multi-physics features of Nmag. The simulations use a self-consistent approach to determine the AMR values, a technique able to explain experimental AMR measurements of real structures. We also show how the spatially varying current distribution affects the computation of the AMR values and found that the uniform current model, sometimes used in the study of AMR effects, is a very inaccurate approximation and can easily lead to qualitatively wrong results.