The magnetic anisotropy of nanometer thin films and of nanosize structures is discussed. Experimental methods for the quantitative determination of magnetic anisotropy are described. Magnetocrystalline, shape, and magnetoeleastic anisotropy contributions are reviewed, and recent examples for the non-bulk-like magnetic anisotropy and of the temperature dependence of both the magnetization and magnetic anisotropy of nanoscale materials are presented. It is shown that film strain and its relaxation give rise to film thickness dependent anisotropy, which can be misinterpreted as a surface anisotropy. The decisive role of the surface anisotropy for adsorbate-induced spin-reorientation transitions (SRT) is elucidated. The application of x-ray magnetic circular dichroism (XMCD) for the determination of magnetic anisotopy of nanosize islands down to the single atom size is presented.