[en] Magnetic nanostructures, such as dots and dot arrays, nanowires, multilayers and nanojunctions, are reviewed and compared with bulk magnets. The emphasis is on the involved physics, but some applications are also outlined, including permanent magnets, soft magnets, magnetic recording media, sensors, and structures and materials for spin electronics. The considered structural length scales range from a few interatomic distances to about one micrometre, bridging the gap between atomic-scale magnetism and the macroscopic magnetism of extended bulk and thin-film magnets. This leads to a rich variety of physical phenomena, differently affecting intrinsic and extrinsic magnetic properties. Some specific phenomena discussed in this review are exchange-spring magnetism, random-anisotropy scaling, narrow-wall and constricted-wall phenomena, Curie temperature changes due to nanostructuring and nanoscale magnetization dynamics. (topical review)