The solution of a large number of estimation problems addressed in statistical signal processing relies on the second-order statistics of a set of multidimensional samples. Indeed, the covariance matrix of the input observations plays a fundamental role in wireless communications and most array processing applications. However, the true covariance information is most often in practice not available, and so must the practical implementation of the optimal solutions necessarily rely on its sample estimate. Unfortunately, the sample covariance matrix is known to be a consistent estimator of the true theoretical covariance matrix when the sample-size is infinitely larger than the dimension of the observations. Despite its unquestionable interest in practice, where the number of samples is usually not much larger (sometimes, even smaller) than the observation dimension, little analytical insight can still be drawn from the broad literature about estimation problems characterized by the availability of a limited number of samples of arbitrarily high dimension. In this talk, we build on results from the spectral analysis of large-dimensional random matrices, or random matrix theory, and propose a class of asymptotic equivalents for certain spectral functions of the covariance matrix describing the solution to different typical estimation problems. In particular, the problems of reduced-rank linear filtering and signal power estimation will be considered. Our results are uniquely based on the sample covariance matrix but can be shown to be strongly consistent for the realistic case of a bounded number of samples per observation-dimension.