Classical regression analysis relates the expectation of a response variable to a linear combination of explanatory variables. In this article, we propose a covariance regression model that parameterizes the covariance matrix of a multivariate response vector as a parsimonious quadratic function of explanatory variables. The approach can be seen as analogous to the mean regression model, and has a representation as a type of random effects model. Parameter estimation for covariance regression is straightforward using either an EM algorithm or a Gibbs sampling scheme. The proposed methodology provides a simple but flexible representation of heteroscedasticity across the levels of an explanatory variable, and can give better-calibrated prediction regions when compared to a homoscedastic model.