The inner walls of the drinking water distribution system (DWDS) are expected to be clean to ensure a safe quality of drinking water. Complex physical, chemical, and biological processes take place when water comes into contact with the pipe surface. This paper describes the impact of leaching different compounds from the water supply pipes into drinking water and subsequent risks. Among these compounds, there are heavy metals. It is necessary to prevent these metals from getting into the DWDS. Those compounds are susceptible to impacting the quality of the water delivered to the population either by leaching dangerous chemicals into water or by enhancing the development of microorganism growth on the pipe surface. The corrosion process of different pipe materials, scale formation mechanisms, and the impact of bacteria formed in corrosion layers are discussed. Water treatment processes and the pipe materials also affect the water composition. Pipe materials act differently in the flowing and stagnation conditions. Moreover, they age differently (e.g., metal-based pipes are subjected to corrosion while polymer-based pipes have a decreased mechanical resistance) and are susceptible to enhanced bacterial film formation. Water distribution pipes are a dynamic environment, therefore, the models that are used must consider the changes that occur over time. Mathematical modeling of the leaching process is complex and includes the description of corrosion development over time, correlated with a model for the biofilm formation and the disinfectants–corrosion products and disinfectants–biofilm interactions. The models used for these processes range from simple longitudinal dispersion models to Monte Carlo simulations and 3D modeling. This review helps to clarify what are the possible sources of compounds responsible for drinking water quality degradation. Additionally, it gives guidance on the measures that are needed to maintain stable and safe drinking water quality.