This paper provides evidence that the variance of returns on common stocks is not constant through time but is related to the volume of shares traded. In other words, returns on stocks are heteroscedastic. The work extends the approaches of Osborne, Granger and Morgenstern, and Clark.' Distributions of returns are known to be leptokurtic. One reason why heteroscedasticity is interesting is that, in principle, a mixture of normal distributions with different variances could account for the leptokurtosis. This paper finds clear evidence of heteroscedasticity. However, some leptokurtosis remains even after the variance has been stabilized by transformation of the data, so that the heteroscedasticity explanation of the departure from normality is incomplete. The plan of this paper is influenced by the following problem. Some specific distribution of returns must be assumed in order to test for heteroscedasticity. Therefore the conclusions drawn about heteroscedasticity will depend on the appropriateness of the model chosen. In the initial tests, heteroscedasticity does seem to be present, but not all of the evidence is in favor of the underlying model, so the conclusions are suspect. The solution to this problem is to use volume to make conditional predictions of return. These predictive tests give a direct answer to the question of whether knowlledge of volume is useful in specifying a return distribution. In the final section of the paper, some of the shortcomings of the basic model are analyzed, and evidence of a positive association between volume and return is presented.