In using the fully relativistic versions of the Embedded Cluster and Screened Korringa-KohnRostoker methods for semi-infinite systems the magnetic properties of single adatoms of Fe and Co on Ir(111) and Pt(111) are studied. It is found that for Pt(111) Fe and Co adatoms are strongly perpendicularly oriented, while on Ir(111) the orientation of the magnetization is only out-of-plane for a Co adatom, for an Fe adatom it is in-plane. For comparison also the so-called band energy parts of the anisotropy energy of a single layer of Fe and Co on these two substrates are shown. The obtained results are compared to recent experimental studies using e.g. the spin-polarized STM technique.PACS numbers: 73.20. At, 72.10.Fk, 75.30.Hx, 73.20.Hb The potential application in non-volatile data storage devices is one of the driving forces behind research into magnetic nanostructures. In state-of-the-art hard disk drives a collection of a few hundred of single-domain particles (grains) are used to hold one bit of information. If materials can be manufactured which exhibit sufficiently large anisotropies and thermal stabilities it may become possible to store one bit in a single grain [1]. Such storage devices will require magnetic structures of precise atomic arrangement, as -if in addition the lateral dimensions of grains are further reduced -the influence of the perimeter atoms becomes increasingly important [2,3] and, as is known, from previous studies the magnetic properties of each atom in a nanostructure are highly influenced by its local environment [1,2,3,4].Using Scanning Tunneling Microscopy structures can be precisely tailored and their magnetic properties determined. In recent Scanning Tunneling Spectroscopy experiments [3,5] it has become possible to measure not only the Lande g-factor of individual atoms but also their magnetic anisotropy. The findings suggest that the anisotropy energy of a single atom may eventually be large enough to use the magnetic state of an atom as a storage unit, pushing the ultimate limit for data storage density even further. Since the magnetic properties of small clusters and single adatoms differ strongly from those of bulk systems and even monolayers -e.g. showing a much enhanced magnetic anisotropy energy -they do not only generate interest for their technological relevance but also from a fundamental point of view.In this paper we present a study of the magnetic moments and the angular dependent band energy part of the magnetic anisotropy energy of single atoms of Fe and Co, which -in order to investigate the influence of different substrates -have been deposited on Pt(111) and Ir(111). The calculations have been performed by means of the Embedded Cluster Method (ECM), a scheme based on the fully relativistic Screened Korringa-Kohn-Rostoker (SKKR) method, in which we can treat impurities embedded into a two-dimensional translationally invariant semi-infinite host. This approach makes use of multiple scattering theory in which the electronic structure of a cluster of embedded atoms...