In this thesis we discuss the realization of a scanning tunneling microscope (STM) operating at temperatures below 20 mK. This is accomplished by attaching the STM to a dilution refrigerator. We can apply high magnetic fields up to 14 T perpendicular and 0.5 T parallel to the sample surface. A full ultra high vacuum (UHV) preparation chamber is attached to the cryostat allowing in situ preparation of the samples to investigate. This setup allows us to investigate low temperature phase transitions in solids and offers the possibility for ultimate energy resolution tunneling spectroscopy measurements on the atomic scale. The concept of realizing the mK-STM is given. A main point of concern was the design of the STM unit. The choice of materials should be well adapted for the use at very low temperatures and in high magnetic fields. A special concern are thermal expansion coefficients and thermal conductivity. In addition, we analyzed the variation of the tip sample distance for a given external excitation for our STM design and two variations of the Besoke-STM by using a one dimensional mass spring model. This gives us the possibility to extract an optimization criterion for our design to limit the response of the tip sample distance to vibrations that are transmitted to the STM. The external damping mechanism, which ensures a low mutual vibration level of the STM as well as the grounding concept for the electronics is explained. To characterize the STM at operating temperatures of 1.5 K, 800 mK and 15 mK we measured topography images of Au(111) and Cu(111) surfaces. This allows us to extract the stability between tip and sample. A statistical analysis of the noise in the tunneling current without activated feedback loop yields a stability of 2.7±0.1 pm in a frequency range f ≈ 0.03 – 25 Hz at 800 mK. To verify if tip and sample are cooled efficiently, we measured the temperature at their positions with additional thermometers. We achieved 17 mK at the tip and 20 mK at the sample position, respectively. The energy resolution of the STM setup was verified by taking tunneling spectra between a superconductor and a normal conducting metal. These spectra could be fitted using the BCS theory of superconductivity. From those ts we can extract the upper limit of the effective temperature of the electrons which is Teff = 87 ± 3 mK for our best measurement. This corresponds to an energy resolution of 3.5 kbTeff = 26 ± 1 µeV. Further, the tunneling conductance between two superconductors was investigated. A sub-gap structure, which can be related to Andreev reflections in asymmetric tunneling junction was observed. In addition, at zero bias voltage a peak due to the Josephson effect can be measured. Side peaks with intensities that scale with this zero bias conductance peak appear most probably due to the interaction of Cooper pairs with the electromagnetical environment in which the junction is placed. Investigating the dependence of the conductance between a superconducting tip and a normal conducting sample on externally applied magnetic fields leads to the appearance of shoulders in the coherence peaks. These shoulders shift linearly with the external magnetic field. Such an effect has been measured for planar tunnel junctions and can be explained with the lifting of the spin degeneracy of the quasi-particle density of states in a superconductor. It was observed for the first time by Meservey, Tedrow and Fulde. The fact, that we are now able to prepare STM tips showing the Meservey-Tedrow-Fulde effect enables us to measure the absolute spin polarization at EF with atomic resolution.