High-entropy alloys (HEAs) are emerging as a new family of alloys with equal/near-equal amounts of constituting elements and outstanding properties. In particular, eutectic high-entropy alloys (EHEAs) with alternate lamella phases possess both high strength and ductility, offering the advantage of conquering the strength–ductility trade-off that could hardly be achieved by conventional alloys. While the mechanical behavior of EHEAs has been widely studied, the corrosion behavior is still not fully understood. Furthermore, the environment-induced degradation could largely decide the service life of EHEA as engineering alloys, and the eutectic structure may have a special influence on the corrosion process. This article systematically reviews the corrosion studies of EHEAs by pointing out the structural features of EHEAs, summarizing the general corrosion issues for EHEAs and identifying the specific corrosion performance of different EHEA systems. It is found that EHEAs feature micro-galvanic corrosion due to their eutectic crystal structure, and such a corrosion mode is further affected by testing time, heat treatment, temperature, and applied potential. All the corrosion-affecting factors are summarized, and future research directions are suggested, aiming at ensuring the wide engineering application of EHEAs with both high strength–ductility and corrosion resistance.