The history of the use of symmetry arguments in probability theory is traced. After a brief consideration of why these did not occur in ancient Greece, the use of symmetry in probability, starting in the 17th century, is considered. Some of the contributions of Bernoulli, Bayes, Laplace, W. E. Johnson, and Bruno de Finetti are described. One important thread here is the progressive move from using symmetry to identify a single, unique probability function to using it instead to narrow the possibilities to a family of candidate functions via the qualitative concept of exchangeability. A number of modern developments are then discussed: partial exchangeability, the sampling of species problem, and Jeffrey conditioning. Finally, the use or misuse of seemingly innocent symmetry assumptions is illustrated, using a number of apparent paradoxes that have been widely discussed.