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The atomic-scale and mesoscopic physics of magnetic nanostructures is reviewed. Emphasis is on the description of magnetic phenomena and properties by analytical models, as contrasted to numerical approaches. Nanostructuring affects the magnetic properties on different length scales, from a few interatomic distances for intrinsic properties such as magnetization and anisotropy to more than 10 nm for extrinsic properties, such as coercivity. The consideration includes static and dynamic mechanisms, as well as nanoscale finite-temperature effects. Some explicitly discussed examples are Curie- temperature changes due to nanostructuring, the effect of narrow and constricted walls, the potential use of magnetic nanodots for finite-temperature quantum computing, and exchange-coupled hard-soft nanocomposites. The temperature dependence of extrinsic properties reflects the atomic-scale static or ‘intrinsic’ temperature dependence of the free-energy barriers and thermally activated dynamic or ‘extrinsic’ jumps over metastable free-energy barriers.