Astronomers working in New York have discovered the first exomoon to be 'seen' from earth.
How they did so is interesting. The various planetary systems discovered around distant stars (known as exo-planets) - and there are several thousand of them at this stage - are not directly visible to astronomers. Instead their presence is detected by the fact that as they pass between us and their star the light from that star seems to dim, and also because gravitational forces can cause tiny movements in the star itself.
the astronomers were observing one such exoplanet - Kepler 1625-b - and noticed what they call deviations and wobbles. They believe that this behaviour can only be explained by the presence of a very large (Neptune-sized) moon.
and the ESA has a good page on how this search is carried out here: https://www.theguardian.com/science/2018/oct/03/astronomers-discover-first-suspected-exomoon-8000-light-years-away?CMP=share_btn_tw