The estimation of covariance matrices from compressive measurements has recently attracted considerable research efforts in various fields of science and engineering. Owing to the small number of observations, the estimation of the covariance matrices is a severely ill-posed problem. This can be overcome by exploiting prior information about the structure of the covariance matrix. This paper presents a class of convex formulations and respective solutions to the high-dimensional covariance matrix estimation problem under compressive measurements, imposing either Toeplitz, sparseness, null-pattern, low rank, or low permuted rank structure on the solution, in addition to positive semi-definiteness. To solve the optimization problems, we introduce the Co-Variance by Augmented Lagrangian Shrinkage Algorithm (CoVALSA), which is an instance of the Split Augmented Lagrangian Shrinkage Algorithm (SALSA). We illustrate the effectiveness of our approach in comparison with state-of-the-art algorithms.