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Recent trends in the emerging field of surface-supported magnetic nanostructures are reviewed. Current strategies for nanostructure synthesis are summarized, followed by a predominantly theoretical description of magnetic phenomena in surface magnetic structures and a review of experimental research in this field. Emphasis is on Fe- or Co-based nanostructures in various low-dimensional geometries, which are studied as model systems to explore the effects of dimensionality, atomic coordination, chemical bonds, alloying and, most importantly, interactions with the supporting substrate on the magnetism. This review also includes a discussion of closely related systems, such as 3d element impurities integrated into organic networks, surface-supported Fe-based molecular magnets, Kondo systems or 4d element nanostructures that exhibit emergent magnetism, thereby bridging the traditional areas of surface science, molecular physics and nanomagnetism.