We introduce a unified approach to testing a variety of rather general null hypotheses that can be formulated in terms of covariances matrices. These include as special cases, for example, testing for equal variances, equal traces, or for elements of the covariance matrix taking certain values. The proposed method only requires very few assumptions and thus promises to be of broad practical use. Two test statistics are defined, and their asymptotic or approximate sampling distributions are derived. In order to improve particularly the small-sample behavior of the resulting tests, two bootstrap-based methods are developed and theoretically justified. Several simulations shed light on the performance of the proposed tests. The analysis of a real data set illustrates the application of the procedures.