Single-ion magnets (SIMs) are mononuclear molecular complexes exhibiting slow relaxation of magnetization. They are currently attracting a lot of interest because of potential applications in spintronics and quantum information processing. However, exploiting SIMs in, e.g. molecule-inorganic hybrid devices requires a fundamental understanding of the effects of molecule-substrate interactions on the SIM magnetic properties. In this review the properties of lanthanide SIMs in the bulk crystalline phase and deposited on surfaces in the (sub) monolayer regime are discussed. As a starting point trivalent lanthanide ions in a ligand field will be described, and the challenges in characterizing the ligand field are illustrated with a focus on several spectroscopic techniques which are able to give direct information on the ligand-field split energy levels. Moreover, the dominant mechanisms of magnetization relaxation in the bulk phase are discussed followed by an overview of SIMs relevant for surface deposition. Further, a short introduction will be given on x-ray absorption spectroscopy, x-ray magnetic circular dichroism and scanning tunneling microscopy. Finally, the recent experiments on surface-deposited SIMs will be reviewed, along with a discussion of future perspectives.