In this thesis, we discuss several dependence related problems in statistics, which covers three topics I worked on during my PhD study. In the first part, we introduce a new concept of robust-equitability for (nonlinear) dependence measures and identify a robust-equitable copula dependence measure (RCD). We also apply RCD in the application of feature ranking and selection. In the second part, we focus on the estimation of high dimensional spatial covariance matrix. Under the block bandable assumption which arises naturally from spatially correlated data, we propose a rate optimal double tapering estimator for estimating the spatial covariance matrix and demonstrate its advantages over the sample covariance matrix. The third part generalizes the former one in the way that it considers the dependence from both nearby and remote distance grid points, and propose a far-near covariance model which could potentially be used to model teleconnection effect in climate research.