This paper presents methods for estimating large covariance matrices by 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 under the condition that (log p)/n → 0, and derive 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 that model and well-conditioned, then the banded approximations produce consistent estimates of the eigenvalues and eigenvectors of the covariance matrix. The results can be extended to smooth versions of banding and to non-Gaussian distributions with sufficiently short tails. A resampling approach is proposed for choosing the banding parameter in practice. The paper also discusses the performance of the banded estimators in simulations and on real data, showing that the optimal banding parameter depends on p, n, and the amount of dependence in the underlying model. The authors conclude that the banded estimators provide consistent estimates of the covariance matrix and its eigenstructure under the given conditions.This paper presents methods for estimating large covariance matrices by 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 under the condition that (log p)/n → 0, and derive 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 that model and well-conditioned, then the banded approximations produce consistent estimates of the eigenvalues and eigenvectors of the covariance matrix. The results can be extended to smooth versions of banding and to non-Gaussian distributions with sufficiently short tails. A resampling approach is proposed for choosing the banding parameter in practice. The paper also discusses the performance of the banded estimators in simulations and on real data, showing that the optimal banding parameter depends on p, n, and the amount of dependence in the underlying model. The authors conclude that the banded estimators provide consistent estimates of the covariance matrix and its eigenstructure under the given conditions.