In this paper, we propose an iterative scheme to find a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. Some strong and weak convergence theorems are obtained and the sufficient and necessary conditions that the iterative scheme converges strongly to a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping are also obtained.