With thousands of known exoplanets, many as small as the Earth, our detection capabilities are beginning to border on that necessary to detect the largest moons found in our solar system. Exomoons would offer new opportunities to understand the origins of planetary systems, as well as potentially playing an important role in the search for life. I will discuss the various methods proposed to identify such objects, the state of our knowledge based on present observations, and the potential for new discoveries via upcoming observations (such as JWST), as well as new methodological developments. An update on the exomoon candidate Kepler-1625b-i will be offered from new observations and tests. Going forward, it is suggested that the statistical validation of exomoons may enter the fray, akin to many of Kepler’s exoplanets, but JWST could present far more compelling detections should it be used for exomoon hunting.