1I/`Oumuamua (aka 1I/2017 U1) is the first, and so far only, macroscopic object passing through the Solar System from interstellar space on a highly unbound orbit to be detected. It had already passed both perihelion and perigee when it was discovered in data from the Pan-STARRS survey on October 19, 2017. However, it was observed extensively using a wide variety of Earth and space based telescopes until it faded below detection limits in early January of 2018. The resulting data have since been analyzed by many investigators and a wide variety of models for the nature of the object have been proposed, including the exotic possibility that it is an artificial construct of extraterrestrial origin. The empirical properties of 1I/`Oumuamua differ in significant ways from those of all previously known (Solar System) small bodies (comets and asteroids), and none of the proposed models appear able to satisfactorily explain all of the available data in a plausible way. `Oumuamua thus represents an intriguing scientific mystery which may well prove intractable for the foreseeable future.