Recently, it has been showed in Castellani-Giuli [13], that the proof of the existence result for quasimonotone Stampacchia variational inequalities developed in Aussel-Hadjisavvas [6] can be adapted to the case of equilibrium problems. This proof was based on KKM techniques. In this paper we define and study the so-called quasi-equilibrium problem, that is an equilibrium problem with a constraint set depending on the current point. Our main contribution consists of an existence result combining fixed point techniques with stability analysis of perturbed equilibrium problems.