In this presentation, we will talk about finite element approximations of nonlinear eigenvalue problems resulting from modeling micro-structures of matter based on density functional theory. We will present a priori and a posteriori error estimates of finite element approximations. In particular,we will show the convergence and complexity of adaptive finite element approximations and report some numerical results of finite element electronic structure calculations that support our theory. This presentation is mainly based on some joint works with Huajie Chen, Xingao Gong, and Lianhua He.