Fresh-cut potatoes (FCP), like other fresh-cut (minimally processed) vegetables, are a convenient but highly perishable product. Unlike most fresh-cut vegetables, which are “ready-to-eat”, FCP must be cooked before consumption. Therefore, in addition to the safety (chemical and microbiological), quality and sensory characteristics of raw FCP, the same requirements should be applied for cooked potatoes. It is known that many factors play a role in meeting all these requirements: (i) selection of cultivars less susceptible to browning; (ii) use of anti-browning and antimicrobial agents and/or certain physical methods against browning and microbial growth; (iii) packaging and cold storage conditions. In recent studies on FCP, scientists have attempted to deepen their knowledge of the mechanisms of browning prevention to better understand changes at the molecular level as well. The main objective of this review is to provide a comprehensive overview of recent research, which aimed at deepening knowledge of the various changes that occur in potatoes during processing, and to develop new approaches that could help improve quality and extend FCP shelf life. It also discusses the effects of subsequent cooking of FCP on sensory and other properties, as well as on chemical constituents.