Covariance matrices play a central role in statistical analysis. A large
collection of fundamental statistical methods require the estimation of
covariance matrices. With the emergence of high dimensional data from modern
technologies, estimating large scale covariance matrices as well as their
inverse is becoming a crucial problem in many fields. In this talk we give
some theories to unveil the precision to which (inverse) covariance matrices
can be estimated and to develop general methodologies for optimal estimation
of the (inverse) covariance matrices under various settings.