K-EUSO (KLYPVE-EUSO) is a planned orbital mission aimed at studying ultra-high energy cosmic rays (UHECRs) by detecting fluorescence and Cherenkov light emitted by extensive air showers in the nocturnal atmosphere of Earth in the ultraviolet (UV) range. The observatory is being developed within the JEM-EUSO collaboration and is planned to be deployed on the International Space Station after 2025 and operated for at least two years. The telescope, consisting of $\sim {10}^5$ independent pixels, will allow a spatial resolution of ∼0.6 km on the ground, and, from a 400 km altitude, it will achieve a large and full sky exposure to sample the highest energy range of the UHECR spectrum. We provide a comprehensive review of the current status of the development of the K-EUSO experiment, paying special attention to its hardware parts and expected performance. We demonstrate how results of the K-EUSO mission can complement the achievements of the existing ground-based experiments and push forward the intriguing studies of ultra-high energy cosmic rays, as well as bring new knowledge about other phenomena manifesting themselves in the atmosphere in the UV range.