Sterile neutrinos at the electron-volt (eV) scale have emerged as an enigmatic frontier in the realm of particle physics and astrophysics. The Karlsruhe Tritium Neutrino (KATRIN) experiment, renowned for its precision to measure the mass of the electron antineutrino with a target sensitivity of 0.2 eV$/c^2$ (90% C.L.), extends its reach to explore the existence and properties of eV-scale sterile neutrinos. In this pursuit, KATRIN seeks to unlock the secrets of these elusive particles through meticulous tritium $\beta$-decay endpoint spectroscopy. By scrutinizing the electron energy spectrum, KATRIN aims to detect or constrain the presence of sterile neutrinos, their mixing with active neutrinos and their mass hierarchy, thereby contributing invaluable insights into neutrino oscillations, cosmology and the broader landscape of particle physics. In 2022, KATRIN reported the most stringent limit on the neutrino mass with $m_\nu < 0.8$ eV$/c^2$ (90% C.L.) based on data acquired during the first two science runs of 2019. This paper summarizes KATRIN's search for an eV sterile neutrino, analyzing $5.24 \times 10^6$ tritium $\beta$-electrons from the first two runs.
Additionally, the sensitivity studies based on five measurement campaigns are presented. KATRIN's enhanced sensitivity has the potential to probe a substantial portion of the sterile neutrino parameter space.