The Hamiltonian Mean-Field (HMF) model is a 1D simplifiedversion of the gravitational Vlasov-Poisson system. I will present two recentworks in collaboration with Mohammed Lemou and Ana Maria Luz. In thefirst one, we proved the nonlinear stability of steady states for this model,using a technique of generalized Schwarz rearrangements. To be stable, thesteady state has to satisfy a criterion. If this criterion is not satisfied, someinstabilities can occur: this is the topic of the second work that I will present.