Magnetic impurities generate a wealth of phenomena on surfaces. On metals, conducting electrons screen the magnetic moment giving rise to the Kondo effect. On superconductors, the Yu-Shiba-Rusinov (YSR) states emerge inside the superconducting gap due to the exchange coupling. In this thesis, we use a scanning tunneling microscope (STM) with a base temperature of 10mK to study magnetic impurities on surfaces. This thesis contains two parts, the interplay between magnetic impurities and superconductivity as well as the tunneling processes between YSR states. Concerning the interplay between magnetic impurities and superconductivity, the single impurity Anderson model (SIAM) offers a consistent picture, in which the impurity substrate coupling is a central parameter of experimental relevance. For YSR phenomena, the mean field (MF) approximation is usually sufficient which gives analytical results for a quantitative interpretation of the experimental data. We tune the coupling by the atomic forces in the junction and quantitatively identify the key role of the coupling in the YSR energy. We further investigate multiple Andreev reflections (MARs) involving a single YSR state, confirming the absence of a spin forbidden family of MARs thereby proving the spin non-degeneracy of the YSR states experimentally. The presence of YSR impurities not only gives rise to peaks in the tunneling spectra, but also influences the superconducting ground state. At a small coupling, the impurity spin is unscreened, which results in a pi phase shift in the Josephson transport. At a large coupling, the impurity spin is screened and the phase is 0. The change of the YSR ground state qualifies as a quantum phase transition (QPT). We observe a significant step in the Josephson current when tuning the coupling across the QPT, which signals the 0-pi transition and the change of the ground state. We then go beyond the MF approximation and include correlation effects in the SIAM to fit the Kondo effect in the magnetic field. We present the scaling between the YSR energy and the Kondo temperature connected by the SIAM. Despite the close relation between the two phenomena, the question remains that to which extent the correlation effects persist in the superconducting state. On this, we show that the correlation effects manifest as an offset current in the YSR measurements. Conventionally, the STM tip only serves as a probe to unravel the sample properties. However, in principle the tip, sample and junction form one quantum system. With the capability to controllably introduce YSR states of desired properties on the tip apex (the YSR-STM), we construct a minimal tunnel junction between two discrete levels, a YSR state on the sample and a YSR state on the tip apex. The tunneling between two YSR states, Shiba-Shiba tunneling, features sharp current peaks, the area of which reveals the relaxation dynamics. We further show that the spin plays an important role in Shiba-Shiba tunneling and we find a behavior consistent with paramagnetic impurities. To conclude, this thesis presents multiple experimental aspects of a single YSR state quantitatively explained by the SIAM, both on the MF level and in the fully correlated situation. The possibility to functionalize the STM tip with a YSR state opens more possibilities, and we show the tunneling between two YSR states as a first application, laying the basis for possible extension of the YSR-STM to more scenarios.