I will discuss non-linear stability results concerning the asymptotic behavior of solutions to classical models arising in kinetic theory. Firstly, I will present classical results on the stability of vacuum for the Vlasov--Poisson system. Later, I will speak about the stability of vacuum for the Vlasov--Poisson system with an unstable trapping potential (joint work with Anibal Velozo). If time permits, I will also present a stability result for the exterior of the Schwarzschild family of black holes for the spherically symmetric Einstein--massless Vlasov system.