Potentiometric sensors are the largest and most commonly used group of electrochemical sensors. Among them, ion-selective electrodes hold a prominent place. Since the end of the last century, their re-development has been observed, which is a consequence of the introduction of solid contact constructions, i.e., electrodes without an internal electrolyte solution. Research carried out in the field of potentiometric sensors primarily focuses on developing new variants of solid contact in order to obtain devices with better analytical parameters, and at the same time cheaper and easier to use, which has been made possible thanks to the achievements of material engineering. This paper presents an overview of new materials used as a solid contact in ion-selective electrodes over the past several years. These are primarily composite and hybrid materials that are a combination of carbon nanomaterials and polymers, as well as those obtained from carbon and polymer nanomaterials in combination with others, such as metal nanoparticles, metal oxides, ionic liquids and many others. Composite materials often have better mechanical, thermal, electrical, optical and chemical properties than the original components. With regard to their use in the construction of ion-selective electrodes, it is particularly important to increase the capacitance and surface area of the material, which makes them more effective in the process of charge transfer between the polymer membrane and the substrate material. This allows to obtain sensors with better analytical and operational parameters. Brief characteristics of electrodes with solid contact, their advantages and disadvantages, as well as research methods used to assess their parameters and analytical usefulness were presented. The work was divided into chapters according to the type of composite material, while the data in the table were arranged according to the type of ion. Selected basic analytical parameters of the obtained electrodes have been collected and summarized in order to better illustrate and compare the achievements that have been described till now in this field of analytical chemistry, which is potentiometry. This comprehensive review is a compendium of knowledge in the research area of functional composite materials and state-of-the-art SC-ISE construction technologies.