This paper studies a class of information transmission processes which we call rumours. The distinctive features of these processes are that the information transmission takes place in such a way that the recipient does not quite know whether or not to believe the information, and that the probability that someone receives the information depends on how many people already have it. Specifically, we study an example in which there is an investment project with returns only known to a few people. These people have a cost of undertaking the project and this cost is private information. The only information the other agents receive is that someone else has invested. They are not told whether that person actually knew the returns, or whether he too was just acting on the basis of his observation of others. They use their optimal Bayesian decision rule to decide whether to invest. We show that, for a wide class of alternative specifications, this decision rule has the property that a positive fraction of those who observe the rumour will not invest. In this sense a rumour cannot mislead everybody. Counter-intuitive comparative statics results are obtained. For example, more information and higher productivity may reduce welfare, while changing the speed with which the rumour spreads has no welfare effect.