Recently (September 2015) Boyajian et al. presented time-series measurements from the Kepler spacecraft of the brightness of a star known as KIC 8462852. This appears to be a rather typical main-sequence F3 class star whose mass, luminosity, and radius are respectively 1.43, 4.68, 1.58 times those of our own Sun. It is seen from Earth in the constellation of Cygnus, at an estimated distance of 1480 light years. However, the photometric data also show bizarre dips of up to 20% in relative brightness that are aperiodic, irregularly shaped, and can last anywhere between 5 and 80 days. Boyajian et al. discussed various potential natural causes for these dips, and found none that could provide a satisfying explanation of the observed behavior, although occultations by a family of massive exocomet or planetesimal fragments produced during a recent break-up event might just be plausible. Subsequently, Wright et al. ignited an internet firestorm, by pointing out that the observed behavior matches strikingly well with previously published predictions of the occulting effects of hypothetical swarms of planet-sized "mega structures" that might be constructed in orbit around a star by some highly advanced alien civilization. While most astronomers consider this to be a very, very low-probability scenario, it does match the observations better than any natural mechanism considered so far, and is thus widely acknowledged to be worthy of further investigation. Here I will present the data available to date, explain why natural mechanisms are problematic, and describe some recent results that further compound the mystery.