A simple way of understanding standards of proof is in terms of degrees of probability. On this account, to prevail in a civil case a claimant need only prove the defendant's liability to a degree above 0.5. For the prosecution to succeed in a criminal case, it needs to prove guilt to a considerably higher degree: 0.95, say. The proof paradoxes are a set of examples, well known to evidence lawyers, that are often taken to suggest that there is something wrong with this probabilistic account of standards of proof. One example is Blue Bus: Mrs. Brown is run down by a bus; 60 percent of the buses that travel along the relevant street are owned by the blue bus company, and 40 percent by the red bus company. The only witness is Mrs. Brown, who is color-blind. Mrs. Brown appears to be able to establish a 0.6 probability that she was run down by a blue bus. Yet the overwhelming intuition is that the 60 percent statistic is not sufficient for Mrs. Brown to prove her case in a civil trial. Thus, the argument goes, proof involves something more than just probability. After introducing other similar examples, this article undertakes a detailed examination of this type of "proof paradox." It focuses on particular analyses of these paradoxes, distinguishing between inferential, moral, and knowledge-based analyses, in the process touching on issues such as the reference class problem and the lottery paradox. The article emphasizes the richness and complexity of the puzzle cases and suggests why they are difficult to resolve.