Abstract: I present some results and open problems concerning the long-time behavior of solutions to collisionless kinetic equations of Vlasov type. These are nonlinear transport equations describing the mean-field evolution of the phase-space distribution function of particles in a plasma. In particular, I discuss how stable dynamics are generated through the interplay between decay mechanisms—phase mixing on the torus and dispersion in Euclidean space— and the nonlinear propagation of regularity.