Written literature on this topic can be traced at least to Aristoteles about 350 BC, and relevant questions are addressed sporadically in classical literature; however few reliable quantitative predictions are available. Currently open problems are becoming recognized (among them a curious connection with billiards theory), and there is activity toward resolving them. I will discuss the current state of that theory, with emphasis on the question of determining when a body whose density exceeds that of a given liquid can be made to float on a surface of that liquid. Much of the new material I will cover is joint with T.I. Vogel.