This paper addresses the problem of estimating a covariance matrix of \( p \) variables from \( n \) observations using either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse covariance matrix. The authors show that these estimates are consistent in the operator norm as long as \( (\log p)/n \to 0 \), and provide explicit rates of convergence. The results are uniform over well-conditioned families of covariance matrices. They also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in this model and well-conditioned, then the banded approximations produce consistent estimates of the eigenvalues and eigenvectors of the covariance matrix. The paper extends these results to smooth versions of banding and to non-Gaussian distributions with short tails. A resampling approach is proposed to choose the banding parameter in practice, and numerical results on simulated and real data are provided to illustrate the performance of the proposed methods.This paper addresses the problem of estimating a covariance matrix of \( p \) variables from \( n \) observations using either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse covariance matrix. The authors show that these estimates are consistent in the operator norm as long as \( (\log p)/n \to 0 \), and provide explicit rates of convergence. The results are uniform over well-conditioned families of covariance matrices. They also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in this model and well-conditioned, then the banded approximations produce consistent estimates of the eigenvalues and eigenvectors of the covariance matrix. The paper extends these results to smooth versions of banding and to non-Gaussian distributions with short tails. A resampling approach is proposed to choose the banding parameter in practice, and numerical results on simulated and real data are provided to illustrate the performance of the proposed methods.