In 2007, astronomers detected a mysterious burst of radio waves, quite unlike anything they had seen before. Not only was it much brighter and shorter than typical observations, but the burst seemed to come from outside of our galaxy, far further away than known radio sources. At the time, astronomers knew of no source that could produce such a signal. Over a decade and more than 60 similar detections later, we are still none the wiser. In this talk, I will present the current state of observations, look at what makes these bursts so puzzling and discuss theories for what in the universe could be producing them.