Magnetic nanostructures are being developed for use in many aspects of our daily life, spanning areas such as data storage, sensing and biomedicine. Whereas patterned nanomagnets are traditionally two-dimensional planar structures, recent work is expanding nanomagnetism into three dimensions; a move triggered by the advance of unconventional synthesis methods and the discovery of new magnetic effects. In three-dimensional nanomagnets more complex magnetic configurations become possible, many with unprecedented properties. Here we review the creation of these structures and their implications for the emergence of new physics, the development of instrumentation and computational methods, and exploitation in numerous applications.