Linear response eigenvalue problems (LREPs) arise from the study of collective motion of many particle systems, such as excited states of electronic structures. Large scale LREPs are challenging due to its non-symmetric structure. In this talk, we will present recent theoretical results on variational principles of LREPs and discuss conjugate gradient-like algorithms. Numerical results of very large LREPs for computing multiple low-lying excitation energies of molecules by the time-dependent density functional theory will be presented. This is a joint work with Ren-cang Li, Dario Rocca and Giulia Galli.