  Covariance matrices of random vectors contain information that is crucial for
modelling. Certain structures and patterns of the covariances (or correlations)
may be used to justify parametric models, e.g., autoregressive models. Until
now, there have been only few approaches for testing such covariance structures
systematically and in a unified way. In the present paper, we propose such a
unified testing procedure, and we will exemplify the approach with a large
variety of covariance structure models. This includes common structures such as
diagonal matrices, Toeplitz matrices, and compound symmetry but also the more
involved autoregressive matrices. We propose hypothesis tests for these
structures, and we use bootstrap techniques for better small-sample
approximation. The structures of the proposed tests invite for adaptations to
other covariance patterns by choosing the hypothesis matrix appropriately. We
prove their correctness for large sample sizes. The proposed methods require
only weak assumptions.
  With the help of a simulation study, we assess the small sample properties of
the tests.
  We also analyze a real data set to illustrate the application of the
procedure.
