In this paper we study the scalar equilibrium problem (EP). We employ variational convergences of bifunctions (lopsided convergence in the maxinf framework, hypo-convergence, and continuous convergence) to study this problem by means of an approximation method. This method allows us to obtain not only existence but also stability results. We introduce a new notion of approximate solution of EPs and study its properties. Then, by coupling this notion of approximate solution and the above-mentioned variational convergences, we introduce new notions of well-posedness for EPs and characterize them. We identify various classes of problems that are well-posed. Finally, by employing the obtained results we prove convergence results of two numerical methods for pseudomonotone bifunctions.