This paper investigates the nature and the presence of bubbles in financial markets. Are bubbles consistent with rationality? If they are, do they, like Ponzi games, require the presence of new players forever? Do they imply impossible events in finite time, such as negative prices? Do they need to go on forever to be rational? Can they have real effects? These are some of the questions asked in the first three sections. The general conclusion is that bubbles, in many markets, are consistent with rationality, that phenomena such as runaway asset prices and market crashes are consistent with rational bubbles. In the last two sections, we consider whether the presence of bubbles in a particular market can be detected statistically. The task is much easier if there are data on both prices and returns. In this case, as shown by Shiller and Singleton, the hypothesis of no bubble implies restrictions on their joint distribution and can be tested. In markets in which returns are difficult to observe, possibly because of a nonpecuniary component, such as gold, the task is more difficult. We consider the use of both "runs tests" and "tail tests" and conclude that they give circumstantial evidence at best.