The present topical review focuses on recent advances concerning an intriguing phenomenon in condensed matter physics, the scattering of conduction electrons at the localized spin of a magnetic impurity: the Kondo effect. Spectroscopic signatures of this effect have been observed in the past by high-resolution photoemission which, however, has the drawback of averaging over a typical surface area of 1mm2. By combining the atomic-scale spatial resolution of the scanning tunneling microscope (STM) with an energy resolution of a few tens of νeV achievable nowadays in scanning tunneling spectroscopy (STS), and by exposing the magnetic adatom to external magnetic fields, our understanding of the interaction of a single magnetic impurity with the conduction electrons of the nonmagnetic host has been considerably deepened. New insight has emerged by taking advantage of quantum size effects in the metallic support and by decoupling the magnetic adatom from the supporting host metal, for instance by embedding it inside a molecule or by separating it by an ultrathin insulating film from the metal surface. In this way, Kondo resonances and Kondo temperatures can be tailored and manipulated by changing the local density of states of the environment. In the weak coupling limit between a Kondo impurity and a superconductor only a convolution of tip and sample DOS is observed while for strongly coupled systems midgap states appear, indicating superconducting pair breaking. Magnetic impurities with co-adsorbed hydrogen on metallic surfaces show pseudo-Kondo resonances owing to very low-energy vibrational excitations detected by inelastic tunneling spectroscopy. One of the most recent achievements in the field has been the clarification of the role of magnetic anisotropy in the Kondo effect for localized spin systems with a spin larger than S = 1/2.