Artículo de publicación ISIThe aim of this work is to study a notion of variational convergence for vector-valued functions. We show that it is suitable for obtaining existence and stability results in convex multiobjective optimization. We obtain various of properties of the variational convergence. We characterize it via the set convergence of epigraphs, coepigraphs, level sets, and some infima. We also characterize it by means of two metrics. We compare it with other notions of convergence for vector-valued functions from the literature and we show that it is more general than most of them. For obtaining the existence and stability results we employ an asymptotic method that has shown to be very useful in optimization theory. In this method we couple the variational convergence with notions of asymptotic analysis (asymptotic cones and functions).