This chapter presents four structured covariance matricesStructured covariance matrices that indicate specific relationships among the multivariate responses. Then, it introduces linear covariance modelsLinear covariance models, which posit a linear relationship between the covariance matrix and a set of known symmetric matrices. To assure the covariance matrix is positive definite, two unconstrained parameterizationsUnconstrained parameterization are considered. In addition, four empirical examples are presented to illustrate the usefulness of structured covariance matricesStructured covariance matrices.