The paper deals with constant false alarm rate (CFAR) detection of multidimensional signals embedded in Gaussian noise with unknown covariance. We attack the problem by resorting to the principle of invariance,which proves a valuable statistical tool for ensuring a priori, namely at the design stage, the CFAR property. In this context, we determine a maximal invariant statistic with respect to a proper group of transformations that leave unaltered the hypothesis-testing problem under study, devise the optimum invariant detector, and show that no uniformly most powerful invariant (UMPI) test exists. Thus, we establish the conditions an invariant detector must fulfill in order to ensure the CFAR property. Finally, we discuss several suboptimal (implementable) invariant receivers and, remarkably, show that the generalized likelihood ratio test (GLRT) detector is a member of this class. The performance analysis, which has been carried out in the presence of a Gaussian signal array, shows that the proposed detectors exhibit a quite acceptable loss with respect to the optimum Neyman-Pearson detector.