The spin-polarized scanning tunnelling microscope (STM) can in principle resolve not only the electronic, but also the magnetic surface structure. We model recent STM measurements achieving magnetic resolution on the atomic scale by a first-principles method. It is shown that the signature of a specific magnetic or non-magnetic STM tip can unambiguously be identified. It is also established that the model of Tersoff and Hamann would yield an electronic as well as magnetic contrast of the surface which is well below STM resolution.Keywords: Density functional calculations, tunneling, magnetic surfaces, manganese * corresponding author: w.hofer@ucl.ac.uk Currently, intensive research is undertaken to understand the complex behavior of spin-systems in thin ferromagnetic (FM) and antiferromagnetic (AFM) metal films. The motivation behind this effort is the ever increasing demand for denser storage of information on magnetic disks, the main storage devices in the computer industry. While traditionally the emphasis was on long range spin-interactions, because these can be influenced by controlled growth of magnetic and non-magnetic multilayers, short range interactions have only recently come into focus. In this range neighboring atoms with different magnetic properties can act as nanomagnets, their magnetic orientation driven by competing exchange interactions between the electrons. The field gained a considerable boost last year by the first demonstration of magnetism on the atomic scale [1]. It is expected that the packing density of information on ultrathin magnetic films can ultimately be brought down to the very level of single atoms. We may therefore not exaggerate by saying that nanomagnetism is today the most promising field within the burgeoning area of nanotechnology.The spin-polarized (SP) scanning tunnelling microscope (STM), which was used to perform the first local measurements of antiferromagnetism on an atomic scale, can also serve as a tool to probe complex spin structures on surfaces, as recently demonstrated [2]. In this case a magnetic tip tunes in on the spin states of electrons on the sample surface and reveals the magnetic, rather than the electronic, structure of the sample. But the current theoretical model, employed to elucidate the magnetic contrast, disagrees with experiments by one order of magnitude. In order to exploit the full potential of this method for quantitative measurements of magnetism on the atomic scale it is necessary to understand in detail how this contrast actually arises.We shall show in this Letter that the missing information is the chemical and electronic structure of the tip. To this end we simulate STM measurements from first principles, employing magnetic and and non-magnetic model tips and the STM code recently introduced (bSCAN [3]). In particular we analyze, whether (i) the corrugation is really substantially enhanced if a tungsten terminated tip is used [1]; and (ii) whether contamination of the tip with surface atoms might change the images. The first point se...