Magnetism in Finite-Sized Systems down to the Single Atom Limit Pietro Gambardella Ecole Polytechnique de Lausanne Switzerland Advances in magnetic memory cells and sensor elements require nanometer scale structures to operate reliably on timescales varying from picoseconds to years. Yet, from a fundamental point of view, even the magnetization of a single atom deposited on a nonmagnetic substrate constitutes a formidable problem for experimentalists and theorists alike. In this talk, I will show that combined synchrotron radiation spectroscopy and scanning tunneling microscopy experiments can be used to track the evolution of the magnetic properties from single atoms to finite-sized particles constructed atom-by-atom by self-assembling processes, yielding unprecedented insight into magnetic systems below the nanometer scale. I will analyze the electronic structure as well as the local spin and orbital moments of individual transition-metal impurities deposited on metal and semiconducting substrates, and observe how the magnetic anisotropy, the type of magnetic order, and the switching dynamics develop in clusters of increasing size as a function of the atomic coordination. Unusual interface effects, unquenched orbital magnetism, giant magnetic anisotropy, and oscillations of the easy magnetization direction in one-dimensional atomic chains constitute an intriguing part of this study. Finally, I will present epitaxial growth methods to fabricate ordered arrays of metal and metal-organic magnetic clusters and give an overview of the challenges that lie ahead in this field.