In this talk I will discuss the problem of global existence and stability of special solutions to kinetic collisional equations that model the dynamics of gas particles. In the first half of the talk I will consider the problem of stability of global Maxwellians for the Landau equations (and Vlasov--Poisson--Landau for the charged case) on a torus and how collisions cause the global Maxwellians to be stable. In the second half I will focus on global problems on the whole space and discuss how the problem of stability is way richer in this case due to the complex interaction of entropic dissipation (due to collisions) and dispersion.