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The growth and magnetism of nanosize Fe clusters on Pt and other metal surfaces are investigated. Fe clusters have been fabricated directly on the substrates by buffer layer assisted growth under ultrahigh vacuum conditions. The mean cluster diameter and the average cluster spacing were controlled by the Fe coverage and the buffer layer thickness. The enhanced magnetic anisotropy of such clusters of diameters between 0.5 and 10 nm with respect to bulk is discussed. Interface anisotropy contributions are compared with direct dipolar cluster-cluster interaction and indirect interactions mediated by the substrate, including preasymptotic ferromagnetic interaction. It is found that this preasymptotic exchange is rather strong in exchange-enhanced substrates, such as Pt, but it decreases rapidly with increasing distance between clusters and becomes negligible for the experimental cluster spacings in this work. Except for clusters that nearly touch each other, the leading interaction contributions are RKKY-type exchange and magnetostatic dipole interactions.