  Since the discovery of a planet transiting its host star in the year 2000,
thousands of additional exoplanets and exoplanet candidates have been detected,
mostly by NASA's Kepler space telescope. Some of them are almost as small as
the Earth's moon. As the solar system is teeming with moons, more than a
hundred of which are in orbit around the eight local planets, and with all of
the local giant planets showing complex ring systems, astronomers have
naturally started to search for moons and rings around exoplanets in the past
few years. We here discuss the principles of the observational methods that
have been proposed to find moons and rings beyond the solar system and we
review the first searches. Though no exomoon or exoring has been unequivocally
validated so far, theoretical and technological requirements are now on the
verge of being mature for such discoveries.
