The covering generalized rough sets are an improvement of traditional rough set model to deal with more complex practical problems which the traditional one cannot handle. It is well known that any generalization of traditional rough set theory should first have practical applied background and two important theoretical issues must be addressed. The first one is to present reasonable definitions of set approximations, and the second one is to develop reasonable algorithms for attributes reduct. The existing covering generalized rough sets, however, mainly pay attention to constructing approximation operators. The ideas of constructing lower approximations are similar but the ideas of constructing upper approximations are different and they all seem to be unreasonable. Furthermore, less effort has been put on the discussion of the applied background and the attributes reduct of covering generalized rough sets. In this paper we concentrate our discussion on the above two issues. We first discuss the applied background of covering generalized rough sets by proposing three kinds of datasets which the traditional rough sets cannot handle and improve the definition of upper approximation for covering generalized rough sets to make it more reasonable than the existing ones. Then we study the attributes reduct with covering generalized rough sets and present an algorithm by using discernibility matrix to compute all the attributes reducts with covering generalized rough sets. With these discussions we can set up a basic foundation of the covering generalized rough set theory and broaden its applications.