The technology of prestressing concrete structures has been a well-established field in structural engineering since the last century. Nowadays, new materials developed for reinforcing concrete structures, such as fibre-reinforced polymers (FRPs), open further possibilities for engineers. FRP tendons are more and more often investigated as an efficient and suitable material for prestressing, and there are already some field applications of this technology. This paper aims to describe the current state of knowledge in the field of prestressing with FRP tendons, with regard to available materials, developed anchorages and undertaken investigations. Researches conducted in areas such as bond strength, ductility, durability, long-term behavior, and high-temperature resistance are described and summarized. Our goal was to provide a summary of what has been achieved so far in the research field of prestressing concrete structures with FRP tendons; however, field applications and standardization issues are not part of this review.