Nanometre scale magnetic particles (`nanoelements' or `nanomagnets') form a rich and rapidly growing new area in condensed matter physics, with many potential applications in data storage technology and magnetic field sensing. This paper reviews an extensive study into the influence of shape on the properties of nanomagnets in the size range 35-500 nm. Elliptical, triangular, square, pentagonal and circular geometries have all been considered. It is shown that the size, thickness and geometric shape of nanomagnets all play a vital role in determining the magnetic properties. The shape, size and thickness of a nanomagnet are shown to be linked to its magnetic properties by two distinct phenomena. The first is called configurational anisotropy and describes the role played by small deviations from uniformity in the magnetization field within the nanostructures, which allow unexpected higher-order anisotropy terms to appear. These anisotropies can often dominate the magnetic properties. The second is the competition which exists between exchange energy and magnetostatic energy. This competition determines whether the nanomagnets exhibit single domain or incoherent magnetization and also controls the non-uniformities in magnetization which lead to configurational anisotropy. Understanding the influence of shape opens the way to designing new nanostructured magnetic materials where the magnetic properties can be tailored to a particular application with a very high degree of precision.